Some actors called it "judging," as in "I judged last week," a verb meaning, "to play a judge on 'Law & Order.' " Donna Hanover, a broadcast journalist and ex-wife of former New York City Mayor Rudolph Giuliani, judged seven times. Humorist Fran Lebowitz donned the robes on 12 occasions.

"To be part of that judge pool, it's an honor," says Stephen McKinley Henderson, who played Judge Marc Kramer on seven episodes over the last 15 years and is also a Tony nominee this year for "Fences."

To...