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Baby Moon Restaurant. John Gambino and Peter Gambino own and operate Baby Moon Restaurant since 1970. Baby Moon cuisine is based on the authentic flavors of Sicily, serving fresh seafood, meat dishes, organic chicken and organic pasta, pastas and pizzas. Baby Moon Restaurant also offers a low-fat and low-carb menu certified by their consulting nutritionist Layne Lieberman, RD, MS, CDN. Baby Moon Restaurant and Pizzeria operates in the Hamptons, NY at 238 Montauk Highway, Westhampton Beach, NY 11978.