A man walks out in pink spandex and a pig nose, and the crowd bursts into laughter that is both stifled and uproarious. The dim, purple lights of The Rockwell cast him in both shadow and spotlight.

On Pico Boulevard just west of Roscoe’s House of Chicken ‘n Waffles is a little club called the Comedy Union, which used to be a dentist’s office and is now L.A.’s premier showcase for black comedians ...