Cream together butter, shortening, sugar, vanilla, and eggs whites.
Add cream cheese and mix; batter may have small lumps at this point.
Add flour, baking soda, cream of tartar and salt; mix until thick dough is formed.
Wrap dough and chill for at least 60 minutes to overnight.
Preheat oven to 375 degrees
Roll half the dough between two pieces of parchment paper with a rolling pin to between ½ inch thick.
Flour a round biscuit cutter, cut out circles and place on baking sheet.
Bake for 8-10 minutes, removing from the oven before tops crack, and before edges become golden. (Use a toothpick to test that the centers are not raw, by inserting the toothpick and verifying that no wet crumbs are stuck to it when it's removed from the cookie.)
Repeat with remaining dough, re-rolling scraps until all the circle have been created and baked.
Mix frosting ingredients: cream cheese, vanilla, powdered sugar, milk, until smooth; color as desired. Wait until cookies are completely cooled to frost.