Preheat the oven to 350 degrees . Using a food processor, blend the sugar and butter until just combined, about 10 seconds. Add the egg whites, orange peel and vanilla and process until just combined, about 10 seconds. Blend in the flour, about 10 seconds.
Line 3 baking sheets with parchment paper. Place 1 teaspoon of the batter onto the cookie sheet and, using the back of a spoon, spread the batter evenly to make a 3-inch-wide circle. Repeat, leaving at least 1 inch between the cookies. Bake 2 sheets at a time, rotating the pans halfway through, until evenly browned, about 12 minutes. Let the cookies cool on the baking sheets. Carefully remove, then serve with the ice cream. Store leftover cookies in an airtight container.