In a large bowl, whisk together the flour, cocoa powder, cream of tartar, baking soda, and salt; set aside.

In the bowl of a stand mixer fitted with a paddle attachment, mix the butter and cream cheese together until smooth and creamy. Add the vegetable oil, 1/2 cup granulated sugar, powdered sugar, egg, vanilla, and food coloring and mix until fully incorporated. Gradually stir dry ingredients into wet ingredients until dough comes together.

Place remaining 3 Tablespoons of sugar into a small bowl. Form dough into balls about the size of a golfball, roll in sugar, then place on lined baking sheet. Repeat process with more dough until you have 8 or so balls spaced 2 inches apart. Using the bottom of a glass, flatten each dough ball to about 1/2" thickness. It's okay if the cookies form a rim on the edges.

Bake cookies for 7-8 minutes or until tops look mostly dry around the edges, yet still glossy in the center--no browning. Remove from the oven and allow to cool on the baking sheet for 3-4 minutes before transferring to a wire rack to cool completely.

In the bowl of a stand mixer fitted with a paddle attachment, mix the butter, cream cheese, yogurt, vanilla, and salt together until fluffy. Gradually add powdered sugar, a cup at a time, until all of the sugar has been added and frosting is smooth.

Spread buttercream onto cooled cookies and immediately add the sprinkles. Serve and enjoy!