In a large mixing bowl, cream butter, 1 cup of sugar and the powdered sugar. Add vegetable oil and eggs and mix well.
In a separate bowl, combine cornstarch, baking soda, cream of tarter, salt, cocoa and flour and stir to mix..
Add to wet ingredients to the dry ingredients and mix until well incorporated.
Roll dough into 1-2 inch balls and place on cookie sheet. I used a large scoop that holds about 1/3 cup of dough
Put ¼ c of sugar into a bowl.
Dampen the bottom of a smooth bottomed glass and press it into the sugar to coat the bottom of the glass. Then press the sugar-coated-glass into the cookie dough balls and lightly press down until your dough begins to flatten and the edges around the glass begin to split.
Bake at 350 degrees F for 8 minutes.
Cool cookies on a cooling rack and then put them in the fridge until you are ready to frost them.
Combine butter, powdered sugar, cocoa and heavy cream in a medium mixing bowl and beat until smooth. Add vanilla and mix until smooth.