Last week, Alexa Ray Joel, on-leave NYU student and daughter of Billy Joel and Christie Brinkley, made her New York City musical debut at the Cutting Room before a standing-room-only crowd that included both parents, her piano teacher, her publicist (whom she shares with Billy), and her voice coach. Alexa sang, played piano (backed by a three-man band), and chatted with the audience. (Sample banter: “My mom said, ‘That song is so good. It’s by Ray Charles, right?’ I said, ‘Uh, no, Mom, it’s by me.’ ”) Brinkley sat close to the stage, bobbed to the beats, and waved her table votive like a lighter at the end, while the more subdued Piano Man kept a low profile in the back. “I’m the luckiest mom in the world,” Brinkley gushed. “She literally came out singing. She would pop out these songs when she was 3, and I would say to Billy, ‘You should record that.’ ” The proud papa always knew his daughter had talent. “She’s pretty much doing this on her own. She had perfect pitch as a baby,” he said. “I guess it’s genetic.” Next: Hillary Haunts Gifford Miller