In a large bowl, use an electric mixer on medium speed to beat together the butter and sugar until light and fluffy.
Add the vanilla and egg, continuing to mix until smooth and creamy. Set aside.
In a small bowl, whisk together the flour, baking soda, cream of tartar and salt until fully combined.
Slowly add the flour mixture to the butter mixture, until a smooth dough has formed. Cover and chill in the refrigerator for at least 30 minutes.
Preheat the oven to 375 degrees and prepare a baking sheet by lining with parchment paper.
Shape Cookie Dough as Preferred:
For circular cookies: Use a 2 Tablespoon cookie scoop and form dough between palms. Place on the prepared baking sheet and press down lightly with palms.
For cut out cookies: Roll dough on a floured surface about ¼ inch thick and cut desired shape with a cookie cutter.
For sprinkle cookies: Prepare using a circular method, dipping the top of the flattened dough ball into a small bowl of sprinkles.
Bake in the preheated oven for 7-8 minutes or until the edges begin to turn golden brown and the top of the cookies are opaque.
Transfer to the cooling rack and cool completely.
Frosting Directions:
In a large bowl, use an electric mixture on low speed to beat together the butter with 1 cup of the confectioner's sugar until light and fluffy.
Add the milk, vanilla and remaining confectioners sugar until a smooth and creamy frosting has formed.
Transfer to a piping bag and frost cooled sugar cookies.
Allow frosting to dry and set completely before stacking.
Notes
Note: To make frosting thicker, add more confectioners sugar. To make frosting thinner, add milk.