On a recent afternoon at a Tex-Mex bar in Brooklyn, Robert Askins was making margaritas, ringing up tabs, and wiping down tables as the lunch crowd thinned out. With a beard and red flannel shirt, he looked like any other bartender in Brooklyn—except he is likely the only one with a play opening on Broadway this spring.

Mr. Askins, 34, is the author of “Hand to God,” a comedy about a Christian puppet ministry in a Texas town that begins performances March 14 at the Booth Theater. Things go a little awry when one of the puppets,...