Sift the flour into a bowl and add the salt, baking soda and cream of tartar. Set aside.
Combine the sugar, powdered sugar, butter, vegetable oil, eggs and vanilla in a large bowl and beat well with an electric mixer. Sift the flour mixture again and gradually combine with the sugar mixture.
Refrigerate the dough for at least 1 hour.
Roll the dough into balls the diameter of a quarter. Place the dough on a lightly greased cookie sheet. Press the bottom of a glass into sugar; then press the glass into a cookie, making it almost flat.
Bake at 350 degrees for 10 to 15 minutes
For the frosting:
Cream the butter, cream cheese and powdered sugar. Add salt, vanilla and half and half until desired spreading consistency. Frost cookies when completely cooled.
Notes
-This recipe calls for rolling the dough into balls, you can also roll out the dough after it is chilled, then cut into shapes. Bake as directed.-I used a cookie scoop (about 3") and made the cookies into large round cookies.