It’s the first day of Pride Month, and Restaurant Row in New York City’s Hell’s Kitchen is electric with anticipation. In a week, Broadway’s finest will descend on Radio City Music Hall for the Tony Awards. Showtunes emanate from the storied piano bar Don’t Tell Mama — a ballad from Jesus Christ Superstar here, a showstopper from Wicked there. Suddenly, I’ve wandered in, a negroni in hand, and I’m belting along. New York City in June is a fusion of the Tony Awards and Pride, a...