Prep: Preheat oven to 375ºF. Line two cookie sheets with parchment paper or silicon mats and set them aside.
Dough: In a large bowl, beat together the softened butter, powder sugar and granulated sugar until light and creamy. Add in the egg and vanilla extract, mix well to combine.
Add flour, cream of tartar, and baking soda. Combine well.
Color and Shape: Split dough into five separate bowls. Add one color to each bowl, work dough until the color is even throughout.
On a floured surface, roll dough to 1/4-inch thick. Cut out egg shaped cookies and place them 2 inches apart on a greased baking sheet. Reroll dough as needed until all dough is used.
Bake: Bake for 6-7 minutes or until edges begin to brown. Allow cookies to cool.
Frosting: In a medium mixing bowl, cream butter and powdered sugar together. Add 1 tablespoons of milk and vanilla. Beat well. If it is too thick, add more milk. If it is too thin, add more powdered sugar.
Transfer frosting to a large ziplocked bag. Cut a small corner off so you can pipe your decorations onto cooled cookies. Serve and enjoy!