I don't know if he can sing, but I'd love to see Mulaney take on J Pierpont Finch! Others I'd like to see on Broadway are Patton Oswalt (he'd be a good Pseudolus too!), Trevor Noah, Demitri Martin, Ali Wong, Ellen Degeneres, and I agree about Mike Birbiglia!
I think there was a similar thread on this topic a few years ago, and I recall someone came up with the idea to have a revival of Spelling Bee where every few months, a new famous comedian comes in to play Panch. I genuinely 100% think this is a brilliant, and even commercially viable idea, and I would absolutely love to see it. (Again - I didn't come up with this idea. I can't seem to find the thread to see who it was, but if the idea was yours, feel free to take credit!)