Cream together 1 cup butter and 1 cup powdered sugar in the bowl of a stand mixer (or use a large bowl and a hand mixer) until light and fluffy.
Stir in 1 cup vegetable oil and 1 cup granulated sugar.
Stir in 2 eggs, one at a time, followed by 1 teaspoon vanilla extract. Scrape down the sides of the bowl and mix again to ensure all of the ingredients are incorporated.
In a large bowl, whisk together 4 cups all-purpose flour, 1 teaspoon baking soda, 1 teaspoon cream of tartar, and 1 teaspoon salt.
Add the flour mixture to the butter mixture with the mixer on low, stirring until just combined.
Roll into 1¼ inch balls. The dough will be soft but you should still be able to roll it into balls right away. If it’s too soft, put it in the refrigerator for an hour before rolling.
Place the balls 2 inches apart on a baking sheet lined with parchment paper or a silicone mat.
Bake for 10–12 minutes, or until the edges are just lightly browned. Let sit for 2 minutes on the baking sheet, then remove and place on a wire rack to cool.
For the frosting:
Place ½ cup butter, 4 cups powdered sugar, ¼ cup milk or heavy cream, and 2 teaspoons vanilla in a large bowl. Mix with a hand mixer until light and fluffy, about 2 minutes. Add more milk to make it thinner, or more powdered sugar to make it thicker, as desired.
Frost the cooled cookies with the frosting. Add sprinkles right after frosting each cookie so they stick.
Notes
Too many cookies? Tap the ".5X" button next to the Ingredient list and that will halve the recipe for you.I like a looser frosting so it creates pretty swirls on top of the cookies. Add a splash more milk if needed to get to your desired consistency. It should be easy to spread. You'll want to add the sprinkles right away so the frosting doesn't harden. If the frosting in the bowl starts to set up as you're frosting the cookies, beat it with the hand mixer for another 30 seconds. Store the cookies in an airtight container at room temperature for up to one week.