The cast of “Sweeney Todd: The Demon Barber of Fleet Street” at New York’s Barrow Street Theatre. (Joan Marcus)

NEW YORK — There are real pies made for the audience in the Tooting Arts Club’s production of “Sweeney Todd.” And if you know anything at all about the musical, that sentence should make you feel equal parts giddy and revolted. The audience walks into a theater that has been fitted out to look like Harrington’s, an ancient, real meat pie shop in London, with dingy walls, cafeteria-style seating and a sign that advertises “Jellied eels no licker.” You’ll be handed beer or wine, and a metal plate of truffle-scented mashed potato and pie, either vegetarian or meat.

If you’ve seen a production of the show, you’re probably wondering about the meat. If you haven’t, well, it has been 38 years since “Sweeney Todd” debuted, so the statute of limitations for spoilers has long expired. “Attend the tale of Sweeney Todd,” as the famous opening number goes: He’s a murderous barber seeking revenge against the crooked judge who stole his wife and daughter. He opens a barbershop above a pie shop whose proprietor, Mrs. Lovett, has fallen into hard times due to the price of meat. But when Todd begins giving very close neck shaves, if you catch my drift, Mrs. Lovett finds a convenient way to dispose of his evidence and solve her restaurant supply-chain problem, soon bringing customers in droves for the best pies in London. And that is why the idea of serving the audience meat pies — really delicious ones, too — is the best joke in theater right now.

“Sweeney Todd”’s pie maker, former White House pastry chef Bill Yosses. (Joan Marcus)

What’s in the pies? As the lyrics to one of the musical’s most famous songs goes, a priest, a poet, a lawyer, and of course, a shepherd. These were not ingredients typically in former White House pastry chef Bill Yosses’s wheelhouse, so when he was invited to make pies for the show, he decided to use chicken — another joke.

“When people eat alligator or rattlesnake” — or, uh, other exotic meats — “they always say, ‘It tastes like chicken,’ ” said Yosses, who dresses up in a Harrington’s costume apron to serve theatergoers most nights. “They’re always asking if they should expect to find fingernails in the pie.”

He made pies for the president, but this opportunity has opened the curtains to a new phase in his career.

“I feel very much a part of the production. But I think my pies feel more a part of the production,” Yosses said. “They’re almost like a character in the play.”

Palpable theater

“Sweeney Todd” isn’t the only New York production whetting audience’s appetites. “Natasha, Pierre & The Great Comet of 1812,” a Broadway musical adaptation of Tolstoy’s “War and Peace” that’s getting plenty of Tony buzz, hands out Russian dumplings. “Waitress,” the Sara Bareilles-penned Broadway musical about Jenna, a baker who is lonely in her marriage, features pies onstage and off, and the first words sung in the show are “Sugar, butter, flour.”

But don’t call it dinner theater, a phrase that conjures up visions of banquet-hall chicken cutlets and 1970s kitsch. All of the food is thematically a part of the show, which keeps it from being a gimmick.

“I don’t know if there is a dinner theater movement, but I would say there is a larger movement for performance to be as enveloping and palpable as possible,” said “The Great Comet” director Rachel Chavkin. “Food is just an extension of that.”

Josh Groban and the cast of “Natasha, Pierre & The Great Comet of 1812.” (Chad Batka)

Just like any other part of a show, food requires a lot of direction and staging. Chavkin and producers taste-tested 20 dumplings at Russian Samovar before they selected the restaurant’s potato-and-onion-filled winner as the one cast members would hand out in the beginning of “The Great Comet.” Baked in a flaky pastry that comes from a family recipe, they are technically pirozhki, but the cast refers to them as pierogi, says Roman Gambourg, a partner in the restaurant and a co-producer of the show. Previous iterations of the show served more traditional pierogi, along with black bread and vodka — all of which were eliminated due to mess. (The vodka, especially, if people overindulged: “I remember one extraordinary show where, very very quietly and respectfully, I watched a woman getting sick,” Chavkin said. “She vomited into her handbag.”)

But when it moved into the Imperial Theatre, “The Great Comet” needed to make sure audience members could eat their dumpling without getting grease on their hands and, in turn, on the seats.

“The idea was to get the dough that doesn’t leave fat particles on your hand and has a handle to raise it out of the box,” Gambourg said. The “handle” is a knot of dough at the top.

Jessie Mueller, center, is Jenna in “Waitress” on Broadway. (Joan Marcus)

It was a similar consideration at “Waitress,” where pies can be purchased for $10, and eaten during the show. Actual waitresses, costumed just like characters in the musical, walk the aisles of the Brooks Atkinson Theatre hawking jarred pies in four flavors, including Key lime and salted caramel, and many audience members find them hard to resist. The theater sells about 1,600 a week, and sometimes, if they are especially popular before the show, the concession stand will call in a new order before intermission. Theatergoers might be influenced by a subtle piece of stagecraft: Shortly before the beginning of every show, pie consultant Stacy Donnelly (yes, that’s her real job title) dumps an excessive amount of cinnamon and nutmeg on a store-bought pie and pops it into a hidden oven in the lobby. It’s not long before the room fills with the scent.

“Waitress” initially wanted to sell slices of pies that look just like Jenna’s, but there was concern that they could be messy. Instead, Donnelly devised a jarred pie, which audience members can stash in their bags if they can’t finish the treat. Her bakery, Cute as Cake, bakes the cakes in the jars, which come with a miniature plastic spoon, selected because it could silently scrape up the last few morsels of Key lime from the sides of the jar without clacking around and distracting other audience members — or worse, the actors.

Audience members “feel like they’re part of the diner,” said Donnelly, who contributed to a forthcoming “Waitress” cookbook. “When you’re having the pies they’re talking about, it makes it a different experience. It’s not just something you grabbed at concessions.”

‘A pretty joyous moment’

Chefs who land a role in these shows are ready for the spotlight.

“Restaurants are kind of theater in themselves, more and more, with an elaborate set and a certain protocol for the service,” said Yosses, who bakes as many as 90 pies per show. Theater is “not so different from the restaurant world. Punctuality is very important.”

Russian Samovar pirozhok used in “Natasha, Pierre & The Great Comet of 1812.” (Closing the Loop Productions)

The jarred pies sold at “Waitress” on Broadway. (Leah Gerstenlauer)

And making and transporting food for the theater is a production in itself. At Russian Samovar, the prep begins every morning around 10, when cooks will start to shape and fill 300 dumplings — 600 on days when there are two shows. Later that day, they’ll be baked, stuffed into tiny cardboard boxes, and transported to the theater in thermal bags that keep them warm, between 15 and 30 minutes before the show begins. The immersive theatrical experience, starring Josh Groban, opens with actors tossing the boxes to guests who want a snack.

“It’s pretty joyous moment,” ensemble member Azudi Onyejekwe said. “We scream, ‘Who wants a dumpling?’ . . . People are sort of like, ‘Me, me, me!’ ”

Because of the show’s unique scenic design — some audience members sit on the stage and the action is spread throughout the theater, even on the mezzanine — every section has an opportunity to get the snack. The cast throws them like they’re using a T-shirt gun at a baseball game. “People are pulling out catches that would make Odell Beckham [Jr.] go ‘Wow,’” Onyejekwe said. But it also tells them what kind of a show “The Great Comet” will be.

“Since the show itself is so much more inclusive in terms of the audience literally being a part of the action every night, there’s something so beautiful about the fact that we’re welcoming them into the space with food,” Onyejekwe said.

“Sweeney Todd” audience members eat their pies and mash in the Harrington’s Pie & Mash Shop recreated inside Barrow Street Theatre. (Joan Marcus)

It’s also a way that theater can engage all of the senses — and compete with other art forms for patrons’ entertainment dollars, especially as Netflix and restaurants spread peoples’ attention spans thin.

“I think it is absolutely a reaction to the increasing virtualization of our world,” Chavkin said. “When people come to a live event, they want it to be live in every sense. They’re looking for as visceral and intimate and personal an experience as possible.”

Yosses thinks that more artists of all types — visual, musical, conceptual — will begin to experiment with serving food for their art. Some, such as performance artist Rirkrit Tiravanija, have long incorporated it into their work.

And Yosses has already seen the effect it has on guests at the Barrow Street Theatre. When they realize they’re about to eat a facsimile of Mrs. Lovett’s pies, they “come in with their libretto already prepared,” he said. Most will sing a line from “A Little Priest.” Others will pepper him with jokes.

“Someone came in, and we served them their pie, and they said, ‘Is this last night’s audience?’ ” Yosses said. “I liked that.”

The pies have been getting rave reviews from the audience. But how do they compare to Mrs. Lovett’s?

“I feel like [Mrs. Lovett] would use more seasoning, and maybe it would be chewier,” said audience member Tim Federle, 36. “But I’m not an expert on human flesh.”