Preheat oven to 375 degrees F.
Cream the sugar, confectioners sugar, butter and oil. Reduce speed of mixer and beat in the egg and vanilla.
In a separate bowl, thoroughly stir together flour, cornstarch, baking soda and cream of tartar. Gradually add flour mixture to butter mixture. Batter should be too soft to roll, but thicker than actual batter. You can bake the cookies immediately or chill the dough until you’re ready. Chilled dough is easier to work with.
When ready to bake, scoop up tablespoons of dough and shape into 1 inch (ish) balls. Place each ball in a cupcake wrapper lined muffin tin and bake for 10-12 minutes or until cookies appear done. Let cool completely. The cookies are very tender, so I froze mine before icing. Ice the frozen cookies. Keep in a covered container.
Makes about 2 ½ dozen cookies