John Mulaney can’t help but play favorites a bit discussing the different parts of his unorthodox weekly talk show Everybody’s Live With John Mulaney. “I have to say,” he told Rolling Stone via Zoom last week, “there’s no part of the show and there’s no part of the week that’s as exciting as when we first rehearse music.”

“Hardcore music nerd” may not be Mulaney’s most well-known attribute, but his level of music geekery is astonishing. (During a recent talk with Rolling Stone, he...