When Michael Cohen was 15, growing up outside of Philadelphia in Cheltenham, Pa., his father told him and his brother it was time to get summer jobs. “My brother went to work in a pet store, because he was interested in fish and reptiles,” Mr. Cohen recalled one sunny afternoon in the garden of East Hampton’s 1770 House. “I said I’m going to work at a restaurant as a busboy. I found a job at a really swanky restaurant, and I got the bug. I’ve had a passion for wine and hospitality ever since.”