Divide the frosting evenly among three medium bowls. Add 1 shade of food coloring to each and stir to make three different colors of frosting.
Transfer each frosting to a piping or resealable plastic bag with one corner snipped. Starting along the outside edge of a cookie, pipe concentric circles of frosting, alternating colors for each circle, until the surface is covered. Drag a skewer from the center of the cookie outward to make lines through the frosting, making sure to clean the tip of the skewer between drags. Continue to drag lines through the circles until a star shape is formed. Repeat the process on each cookie.