Preheat your oven to 350°F. In a medium bowl, mix the flour, baking soda, baking powder, and cream of tartar; set aside.
Using a stand mixer or hand mixer, beat together the butter, sugar, and 3 tablespoons powdered sugar until fluffy, then mix in the oil. Add the egg and vanilla, beating until combined.
Gradually add the dry mixture into the butter mixture until just blended and a soft dough forms.
Portion the dough into 2 ½” balls. Place them on a parchment-lined cookie sheet and flatten to form a 4-5” cookie. You should be able to make 6 very large cookies.
Arrange the cookies on a pan about 2” apart and bake for 9-12 minutes at 350°F. Allow to cool completely on a cookie rack before icing.
For the frosting, beat together the butter and powdered sugar until smooth. Add the milk and vanilla, blending to your desired consistency, adjusting with more milk or sugar as needed. Mix in a few drops of pink food coloring until you achieve the desired shade.
Frost the cooled cookies and store them in the refrigerator.
Notes
This recipe makes 6 large cookies which is about 12 servings. Depending on the size of your baking sheet, these cookies might need to be baked in batches.