Preheat oven to 375ºF.
In a mixing bowl, combine butter and sugars. Beat until creamy. Add egg and vanilla. Mix well.
Add flour, cream of tartar, and baking soda. Combine well.
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.
Bake 6-7 minutes or until edges begin to brown. Allow cookies to cool.
To make 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 ziplock bag. Cut a small corner off so you can pipe your decorations onto cooled cookies.