Make the cookies. In a small bowl, whisk together the flour, cornstarch, baking powder, salt, and baking soda. Set aside.
In a large bowl, cream together the softened butter and confectioner's sugar until light and fluffy, about 2-3 minutes. Add the sourdough discard, egg, sour cream, and vanilla extract, and mix until smooth.
Add the dry ingredients to the wet ingredients, and mix until combined. Transfer the dough to a piece of plastic wrap and shape into a disk. Chill for at least 1 hour.
Preheat the oven to 375℉ (190℃) and line your baking sheets with parchment paper. Set aside.
Divide the disk of dough into 2 halves (this helps prevent overworking the dough). Working with one half at a time, roll the dough out to ¼-inch thickness, then cut out your shapes with the cookie cutters. Transfer the cut cookies to the parchment paper-lined baking sheet, leaving at least 1 inch between each cookie. Gather the dough scraps, then re-roll and cut your cookies, repeating until you've used all the dough.*
Bake for 8-9 minutes until the cookies are set, and just starting to lightly brown on the bottom. Let cool for a few minutes on the baking sheet, then transfer to a cooling rack to cool fully.
Make the frosting. Once the cookies are fully cooled, cream the butter until light and fluffy, 2-3 minutes, then slowly mix in the confectioner's sugar. Add the vanilla extract and milk, mixing until fully smooth and incorporated. Add more milk if you feel that the frosting is too thick. If using food coloring, add a few drops and mix until you reach your desired color.
Spread the cookies with a layer of buttercream frosting, add sprinkles (optional), and serve!
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Notes
*The first roll will be the best. You'll find that with subsequent rolls (the second and third time), the dough is thicker and those baked cookies will be thicker. They're still delicious, but you'll get the "best" shape from the cookies in the first roll of dough. Get as many as you can out if it with each roll!To make these cookies with active sourdough starter, replace the sourdough discard with active starter (200g). You do not need to change any other ingredients in the recipe. 1 serving = 1 cookie.