Preheat oven to 375 degrees. Line two cookie sheets with parchment paper. Set aside.
In a large mixing bowl, beat together white sugar, powdered sugar, butter and oil until smooth.
Add eggs and vanilla; mix well.
In a medium mixing bowl, stir together flour, baking soda, and cream of tartar. Add a little at a time to wet ingredients. Do not over mix. Stir in the candy pieces, if using.
Using a 1/2 ounce cookie scoop (or by rounded teaspoonsful) shape cookies and place on prepared baking pans. Sprinkle each dough mound with a little bit of white granulated sugar.
Bake for 8-10 minutes, just until edges of cookies turn a very light golden brown. The cookies will be pale and may appear underdone. I used two different ovens and 9 minutes in both of them was the perfect amount of time. I am at 4500 feet above sea level, so cooking times may vary at other altitudes.
Store cookies in airtight containers for up to 5 days.