Heat oven to 350°F. Knead flour into dough. On a floured surface, roll dough to 1/4-inch thickness. Cut out egg-shaped cookies using a 3-inch cookie cutter. Place on a baking sheet and bake 20 minutes, until lightly browned. Cool 5 minutes, transfer to wire rack, and cool completely. Reroll scraps to bake more cookies.
In a bowl, with mixer on low speed, beat sugar, egg whites, and water until combined. Increase mixer speed to high, and beat for 5 minutes, until stiff peaks form. Thin icing with water 1 teaspoon at a time, until it's the consistency of thick cream. Using the tip of your index finger, ice cookies. Let dry completely.
Cut a new kitchen sponge into 1-1/2-inch pieces. Dampen pieces and squeeze a few drops of icing color onto a sponge piece and pat tops of the cookies. Repeat with other colors and sponge pieces. Let dry completely.