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Step 1
Let the butter soften at room temperature in the bowl of a stand mixer. When soft, beat until billowy and light. Add the sugar, and mix again until well integrated. Add the eggs and milk, and mix until uniform in color. Combine the flour, baking powder, and salt in a separate bowl, then gradually stir into the butter mixture, either by hand, or with the mixer on low. Divide the dough into four, shape into flat disks, wrap in parchment or plastic, and refrigerate for an hour or two.
Step 2
When you're ready to bake, preheat the oven to 350F. Roll out the dough onto a well-floured (or powdered sugared) countertop to 1/4-inch thick. Cut into desired shapes and arrange on parchment (I reuse the parchment I wrapped the dough in earlier), leaving 1/2-inch between cookies. Chill again in the refrigerator (or freezer) for a few minutes (optional, but the shapes end up with more definition if you do). Bake for 8-10 minutes, until toasty around the edges. Rotate once along the way if your oven has hot-spots. Cool on a rack completely, and proceed with icing.
Step 3
To make the icing, whisk together the powdered sugar and egg whites* until smooth and silky. It takes a few minutes. Reserve a bit of this mixture if you like a more opaque icing. You can thin with water from here depending on your preference. You can also add any coloring you like at this point. Use a brush to paint the icing on the cookies, allow to dry completely, store in an airtight container for up to a week.