Preheat the oven to 350°F. Line two baking sheets with parchment paper and set aside.
Add the softened butter and granulated sugar to the bowl. Cream until smooth.
Add the oil, egg, vanilla extract, and almond extract. Mix until fully incorporated, scraping the bowl to ensure everything is evenly combined.
Add the flour, cream of tartar, salt, and baking powder. Mix thoroughly and scrape down the sides and bottom of the bowl.
Use a large cookie scoop to divide the dough into 8 equal portions. Bake for 10 to 12 minutes, until the centers are set and the edges are just firm. Let the cookies cool completely on the baking sheets.
For the frosting, add the softened butter and powdered sugar to a stand mixer. Mix on low speed until combined.
Add the vanilla extract, almond extract, milk or cream, and pink food coloring. Mix on low until incorporated, then increase to medium speed and beat for 1 to 2 minutes until smooth and creamy.
Spread the frosting evenly over the cooled cookies and serve.