If someone were to build a machine that could create the perfect comedian for late-night television, it might spit someone out like John Mulaney. Born in Chicago in 1982, one of the first glimmers he had of his eventual hunger for the limelight was watching Ricky Ricardo perform with his band on reruns of “I Love Lucy” and both embracing and mocking that old-world sense of showmanship has been part of his charm ever since.

During his youth, Mulaney would also spend time watching archived episodes of “The Tonight Show with Johnny Carson” at the Museum of Broadcast Communications, likely finding inspiration in Carson’s mix of childlike playfulness and debonaire affect. Whereas most stand-ups of his generation perform in casual threads, Mulaney is known for regularly donning a suit in both his specials and appearances, perhaps a nod to Carson or his own way of recognizing the importance of formality...