In a large mixing bowl, beat butter on high speed until light and fluffy. Add sugar and beat until combined.
Add eggs, cream of tartar, baking soda, peppermint, and salt to the butter mixture and beat until combined.
Beat in the flour and press the mixture into a 9 by 13 pan. Bake for 23-25 minutes until dough is light brown. Allow to cool.
In a clean bowl, beat butter on high speed until light and fluffy. Add powdered sugar and peppermint extract to the butter and beat until combined. Add milk as needed to the mixture to make it spreadable.