Preheat the oven to 350° and line a cookie sheet with parchment paper.
In a small bowl, stir together the flour, baking soda, baking powder, and salt.
In a separate larger mixing bowl, cream butter and sugar with an electric mixer until light and fluffy. Add in the egg and vanilla extract and mix to combine.
Gradually add the dry ingredients to the bowl of wet ingredients, about 1 cup at a time, mixing as you add. Divide the cookie dough into 4 bowls and add about 2-3 drops of the food coloring to each one, mixing until you have 4 different colors of cookie dough.
Scoop about 1 teaspoon of dough, roll it into a ball, and place it on the cookie sheet leaving about 1 1/2 inches between each dough ball.
Bake in the preheated oven for 8-10 minutes or until the edges just start to turn a light golden brown. Let the cookies cool on the pan for a few minutes before transferring them to a wire rack to cool completely.
To Make The Frosting:
In a large mixing bowl, beat together 2 sticks of butter and 3 cups of powdered sugar.
Add the remaining 3 cups of powdered sugar and continue mixing.
Add the vanilla extract and mix until the frosting becomes light and fluffy, about 3 minutes.
To Make The Sandwich Cookies:
Scoop frosting into a piping bag.
Pipe the frosting on the bottom side of one cookie.
Place another cookie of the same color on top of the frosting (so the bottom side of that cookie also faces inward).
Gently press the two cookies together so they stick together, but not enough to squeeze the frosting out the sides.