Beat 1 egg with 1 tablespoon of water to make an egg wash. Line 2 baking sheets with parchment. If you have turned off the oven, preheat it now to 350ºF.
For small cookies, cut the logs into 1 1/2-inch pieces. Brush with egg wash and sprinkle with colored sprinkles and bake, or glaze (see recipe below) with icing and top with sprinkles after baking.
For X cookies, cut into 2 1/2- to 3-inch pieces. With a paring knife, make a slit on both ends of each piece and open them out slightly to form an X. Brush with egg wash and sprinkle with colored sprinkles and bake, or glaze with icing and top with sprinkles after baking.
For curved cookies, cut into 2 1/2- to 3-inch pieces. With a paring knife, make 3 or 4 1/4-inch cuts on 1 side of each cookie, being careful not to cut all the way through. Curve them into a crescent shape. Brush with egg wash and sprinkle with colored sprinkles and bake, or glaze with icing and top with sprinkles after baking.
Bake for 18 to 20 minutes, or until light golden brown.
For the icing: Whisk together 2 egg whites and 2 1/2 cups of confectioner’s sugar. Pour into a shallow dish. Carefully dip the tops of the cookies in the icing and set them on a rack. Sprinkle with colored sprinkles and leave until set.
Store between sheets of wax paper in a tin with a tight-fitting lid for up to 1 week.