Preheat oven to 350°F. Line baking sheets with parchment paper.
In a stand mixer fitted with the paddle attachment, cream together butter, granulated sugar, and confectioners' sugar until just combined.
Drizzle in vegetable oil and add eggs; mix well until fully incorporated.
Whisk together flour, cream of tartar, and kosher salt in a separate bowl.
Slowly add the dry flour mixture to the wet ingredients and mix until the dough pulls away from the sides of the bowl.
Using a 2-tablespoon scoop, drop dough balls onto the prepared baking sheets.
Roll each dough ball smooth and round, then press each with the bottom of a wet glass dipped in granulated sugar, flattening the center slightly.
Bake for 7 to 9 minutes until edges just start to brown and centers no longer appear wet. Remove and let cool.
For the frosting, mix butter, sour cream, and salt in a clean bowl until fully combined.
Gradually add confectioners' sugar about 0.5 cups at a time until the frosting reaches desired thickness.
Add a few drops of red food coloring and mix until frosting turns a soft pink.
Spread the pink frosting onto cooled cookies using an offset spatula or butter knife.
Notes
Do not overbake cookies to keep them soft and chewy. Adjust frosting sugar for desired consistency. The frosting pairs perfectly with the delicate flavor of the cookies.