1. Preheat oven to 350°F. Line a baking sheet with parchment paper. Set aside.
2. In the bowl of your stand mixer fitted with the paddle attachment, mix together the butter, oil and both sugars until smooth on medium speed. Beat in the eggs, vanilla, baking soda, cream of tartar and salt and mix for 1 minute, scraping the sides of the bowl as necessary.
3. Turn the mixer to low and add in flour, mixing until combined.
4. Using a large cookie scoop (3 tablespoons) dough onto the lined baking sheet. Using the bottom of a cup, press the cookies down until the are flat in the middle. Bake for 10-12 minutes until edges just start to golden.
5. Allow to cool for 3 minutes on baking sheet and transfer to wire rack to finish cooling.

Frosting

1. In the bowl of your stand mixer fitted with the paddle attachment mix the butter on medium speed until smooth. Turn the mixer to low and slowly add in the powdered sugar. Once incorporated, add the vanilla, milk, and food coloring (if desired) into the bowl and turn the mixer up to medium. Mix for 2 minutes until fluffy, scraping the sides of the bowl as necessary.
2. Spread 1 tablespoon(ish) of the frosting on each cookie.