Preheat oven to 350 degrees. Line cookie sheets with parchment paper; set aside.
In a large bowl, combine cake mix, egg, oil, and buttermilk. Beat with an electric mixer on medium speed until smooth (dough will be stiff).
Using a cookie scoop, shape dough into 1-inch balls. Drop onto prepared baking sheets about 2-inches apart. With the bottom of a glass dipped in flour, flatten to 1/4 inch thickness. Top each with sprinkles.
With the bottom of a glass or cup dipped in flour, flatten to 1/4 inch thickness. Top each with sprinkles.
Bake for 8 to 10 minutes or until firm and the edges are lightly browned. Transfer to wire racks to cool completely.
Divide frosting among 3 small bowls. Tint frosting with food coloring: yellow-2 drops, blue-1 drop, and pink-1 drop of red.
Frost the bottoms of half of the cookies. Top with remaining cookies.
Cook's Note: Recipe yields 30 individual cookies or 15 sandwich cookies. Alternately, you could bake all of the cookies without sprinkles, cool, tint frosting, frost cookies, and top with sprinkles.