As the nation’s founding fathers strut and fret upon the stage of Broadway’s Richard Rodgers Theatre, Jason Bassett plays an unseen role, perched on a snug triangular wooden platform about 10 feet above them.

While the cast of the smash-hit hip-hop musical “Hamilton” weaves through fast-paced raps and intricate wordplay, he follows them word for word, calling the show’s 856 lighting cues and 40 set cues with rapid-fire, split-second timing.

But...