He paused, presumably to consider that this was not the kind of memory he should share with a reporter. His trajectory from pizzeria to late-night host was too improbable to keep to himself, though. Atamanuik is from Topsfield, near Boston, but after his parents divorced when he was 6, he moved to the decidedly less affluent Chelsea, Mass. He lived with his mother in a duplex they shared with her parents, who introduced him to “Your Show of Shows” and Abbott and Costello. His mother taught at Emerson College, where he took a degree in film theory before moving to Los Angeles for a brief stint in the internet division of the Jim Henson Company. Then he went to New York and worked at John’s while he took improv classes. His is a recognizable story of dislocated class — a person who grew up with artistic ambitions but graduated into circumstances that required him to work.

As Trump, his tagline at the end of the cold open for each show is “I’m the president. Can you believe it?” Atamanuik carries that can-you-believe-it attitude around with him. Improv comedy teaches you many things, and one of them is how few talented people actually make it. The basements of New York are full of talented people. Atamanuik seems aware of this reality and how he might go back to it at any time. This precariousness fuels his determination to capitalize on his opportunity.

“I want mine to be realer than his,” Atamanuik said of the relationship between his performance and Trump’s. “By that, I am taking him. I want to make it where people go, ‘Anthony first, Trump second.’ ” This plan, undertaken in 22-minute increments on basic cable, might charitably be called ambitious. Yet what Atamanuik calls his “grand vision” is not without precedent. Dana Carvey did it on “Saturday Night Live,” when he persuaded a generation that George H. W. Bush routinely said “not gonna do it” and “wouldn’t be prudent” — two catchphrases the president didn’t actually use. And The Onion’s hard-partying, earnestly sleazy Joe Biden became the template for parodies of the former vice president, a fictional version on which subsequent fictions were based.

There is also the matter of how many times Atamanuik’s Trump has prefigured the actual president’s behavior. Celebrating House passage of the American Health Care Act on May 4 at the White House, the real Trump said to a group of Republican lawmakers, “I’m president. Can you believe it?” When Mario Cantone appeared on “The President Show” as Anthony Scaramucci, Atamanuik introduced him as “my favorite new team member, who I will eventually betray.” Four days later, Scaramucci was fired. Atamanuik’s Trump looked at the solar eclipse without glasses and predicted that North Korea’s threats against Guam would promote tourism. Trump did both.

It still seems credulous to imagine that Atamanuik could ever eclipse Trump. For one thing, Trump’s entertainment-industry résumé is much longer. He was doing film and television cameos while Atamanuik was still in college, appearing in “Home Alone 2,” “Sex and the City” and a seemingly endless number of late-’90s sitcoms. He had his own television show for more than a decade in “The Apprentice” and then “The Celebrity Apprentice,” each of which ran in prime time on NBC. Trump was a higher order of celebrity than Atamanuik even before his presidency. But watching the impressionist talk about his plans for his show in the pizzeria where he didn’t have to work anymore, I wanted to believe.

“Maybe we rob it,” Atamanuik said, referring to the sheer mass of Trump’s public image. “Maybe like a binary star, we’re slowly pulling gas from that star. We’re building ours up, and maybe we’ll deplete that one.” He continued: “That was the idea for a long time: to be the anti-Trump, to be the exact opposite, but by embodying him.”

It sounds like a fantasy, and it is. There can be no anti-Trump, no single star of competing mass; that may be the very phenomenon that won him the election. Atamanuik’s dream is the kind of Trumpian ambition that takes your own significance as given and works from there. Still, it is the wild idea that makes sense the morning after you shoot your show, having lunch in the restaurant where you used to work. That’s what you do when you get your big break: You imagine breaking bigger.