Cream together butter, sifted powdered sugar, egg, and vanilla.
Set aside.
In a seperate bowl mix flour, baking soda, cream of tartar and salt.
Slowly mix flour mixture into the butter mixture.
Form into a ball, flatten and wrap in saran wrap.
Put into fridge to chill for 30 minutes.
After dough is chilled remove from fridge.
Preheat oven to 400 degrees.
Sprinkle a pastry cloth or clean counter top with a mixture of powdered sugar and flour. You can also cover your rolling pin with flour/sugar mixture.
Roll dough out to 1/8 to 1/4 inch thick.
Cut out cookies and place on ungreased cookie sheet.
Bake for 6 minutes at 400 degrees.
Remove from oven and place cookies onto cooling rack.
Make the frosting.