Preheat the oven to 350 degrees F.
In a large mixing bowl, cream together the butter, oil, granulated sugar, powdered sugar, vanilla and eggs. Add the flour, baking soda, cream of tartar and salt and mix well. Cover and refrigerate the dough for 1 hour.
Using a cookie scoop, drop balls of dough onto ungreased, unlined baking sheets. Smear a dab of butter all over the bottom of a glass, then dip the glass in granulated sugar. Use the glass to flatten each ball of dough, dipping again into the sugar each time. Then sprinkle blue sanding sugar (about 1/2 teaspoon per cookie) over half of the cookies, and purple sanding sugar over the other half.
Bake until the cookies are just barely turning brown, 9 to 11 minutes. Don't overbrown! Allow to cool before eating. The cookies are ultra crumbly!