Preheat the oven to 350 degrees F. Line baking sheets with parchment paper.
To make the cookies, in a medium bowl, whisk together the flour, cocoa, baking soda, baking powder, cream of tartar, and salt.
In a large bowl, use an electric mixer to cream together the butter, granulated sugar and powdered sugar until light and creamy, about 3 minutes. Add the oil, eggs and vanilla and mix until well-combined.
Add the dry ingredients to the sugar/butter mixture and mix until combined. The dough will be stiff, but shouldn't be dry.
Form the dough into 2-inch balls (about the size of a golf ball). Place the cookies several inches apart on the prepared baking sheets.
Place about 1/3 cup granulated sugar in a shallow dish. Dip a flat-bottomed glass into the sugar (spray the bottom of the glass with non-stick cooking spray so the sugar adheres, if needed). Press the glass onto a cookie, flattening it to about 1/4-inch thick. Repeat with each cookie.
Bake for 8-9 minutes, until the cookies are just set. Do not over bake. Cool for a few minutes before transferring to a wire rack to cool completely.
For the frosting, with an electric mixer, cream together the butter and 1 cup of the powdered sugar until very light and fluffy. Add the remaining powered sugar, sifted cocoa powder, heavy cream, and vanilla. Mix for 2-3 minutes until very thick and creamy.
Spread the frosting evenly on the cookies. Store, tightly covered, in the refrigerator for 3-4 days.