Until July 2013, Aimee Stephens’s boss had only ever known her as a man. That month she pulled him aside at R.G. & G.R. Harris Funeral Homes in Garden City, Mich., where she worked as a funeral director, and handed him a letter that explained everything: how she’d felt imprisoned in her body; how, with help from her wife and her therapist, she’d “decided to become the person that my mind already is”; and how she’d soon “return to work as my true self,” dressed “in appropriate business attire.”

Stephens offered to answer any questions her boss, Thomas Rost, might...