Preheat oven to 350F. Line cookie sheets with parchment paper; set aside.
Combine the cake mix, egg, oil, and buttermilk in a large bowl. Beat with an electric mixer on medium speed until smooth. Shape 1-inch mounds of batter into balls and flatten slightly. Place about 2 inches apart on the prepared cookie sheets. Top with sprinkles.
Bake until cookies are golden and firm. 8 to 10 minutes. Cool on a wire rack.
Divide the frosting among three bowls. Use a few drops of food coloring to tint frosting the desired shades of pink, blue, and yellow. Sandwhich the cookies together with frosting.