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Step 1
Preheat oven to 350°F. Line two cookie sheet with silicone baking mats or parchment paper.
Step 2
Cream butter and sugar in the bowl of a stand mixer fitted with the paddle attachment (or a large bowl with a hand mixer). Cream until the mixture is fluffy, about one minute.
Step 3
Add the egg, egg yolk, and vanilla and mix until smooth, then mix in baking soda, cream of tartar, and salt. Add flour and mix until cookie dough forms.
Step 4
Scoop 2 tablespoon sized balls 2-inches apart on cookie sheet. Bake for 11-15 minutes, or until the bottoms are just starting get golden and the top is no longer glossy. Cool at least 10 minutes on cookie sheet before removing. Cool completely before frosting.
Step 5
Mix butter in a large bowl with a hand mixer until smooth, then slowly mix in the powdered sugar a bit at a time. The mixture will be very crumbly. Add the vanilla and mix, then mix in the salt and milk. Beat until smooth and creamy.
Step 6
Tint with food coloring, if desired.
Step 7
Frost cooled cookies. Store loosely covered.
Step 8
Scoop balls of cookie dough and roll them in sugar or cinnamon sugar before baking. Skip frosting. OR
Step 9
Add 1 cup of sprinkles to the dough before scooping. Skip frosting.