1Heat oven to 375°F. Stir cookie mix, butter, flour and egg in medium bowl until dough forms. Divide dough evenly among 3 bowls; tint each dough by stirring in a few drops of desired food color.
2Shape 1/3 cup of each color of dough into a rope about 5 inches long and 1 inch in diameter. Place ropes side by side and a little more than 1/4 inch apart on floured surface; roll until 1/4 inch thick. Cut with 2- to 2 1/2-inch egg-shaped cookie cutter so each cookie has 3 colors. Sprinkle with sugar. Place 2 inches apart on ungreased cookie sheet. Repeat with remaining dough. (When rerolling dough scraps, carefully lay matching colors together.)
3Bake 7 to 9 minutes or until edges are light golden brown. Cool 1 minute; remove from cookie sheet to wire rack. Cool completely, about 30 minutes. Spread frosting on bottoms of half the cookies. Top with remaining cookies.