Preheat oven to 325 degrees.
In a large bowl, cream together butter, powdered sugar, and 1 cup granulated sugar with electric mixer.
Beat in one egg at a time until light and fluffy.
Add oil, vanilla, and grated lemon peel. Beat well.
In a separate bowl, combine flour, cream of tartar, baking soda and salt. Stir well. Add slowly to the creamed mixture.
Chill until cookie dough is easy to handle.
Roll small balls of dough with your hands and place on a baking sheet lined with parchment paper. Flatten each cookie with the bottom of a glass dipped in sugar.
Bake at 325 for 8-10 minutes. Cool on a wire rack.