Holiday dining at the 1770 House in East Hampton this weekend will include a four-course prix fixe tasting menu at $110 per person, plus tax and gratuity. In Sag Harbor, Lulu Kitchen and Bar will host a New Year’s Eve celebration that includes an early, 5:30 p.m., seating and a later second seating at 8 that comes with a glass of champagne, party favors, and a four-course prix fixe.