Preheat oven to 350°F. Line baking sheets with parchment paper. Set aside.
In the bowl of a stand mixer, combine butter, granulated sugar, and confectioners' sugar. Mix on medium speed until just combined.
Drizzle in oil and add eggs. Mix well.
In a separate bowl, whisk together flour, cream of tartar, and salt.
Slowly add the flour mixture to the butter mixture. Allow the mixer to combine until the dough pulls away from the sides of the mixer bowl.
Using a 2-tablespoon scoop, drop the golf-ball size portions of dough onto a lined baking sheet.
Roll each cookie dough ball until it is smooth and round.
Using the bottom of a wet kitchen glass that has been pressed into granulated sugar, press down on each cookie dough ball until the center of the cookie is pressed flatter than the edges. Repeat this on all cookies.
Bake for 7-9 minutes, or until edges are just slightly starting to brown and the center no longer appears wet. Allow the cookies to cool as you make the frosting.
NOTE: It is important to NOT over-bake the cookies. Be sure to check your cookies at 7 minutes. Every oven is different and overbaking can result in a tough, crumbly, tasteless cookie. Let your first batch be a test batch so you can ensure the remaining cookies are perfect and soft!