At the peak of a recent Friday night, the wait staff at Dressing Room in Westport is doing what it always does: expertly winding through packed rooms of white cloth-covered tables balancing trays crammed with locally sourced food, from buckets of bread, bowls of house-specialty “Use a Spoon” chopped salads, ribs, lemon-ricotta ravioli big portions of gooey desserts that most diners seem disinclined to share.