Sift together the flour and baking powder. Set aside.
In a mixer with a paddle attachment, cream the butter, sugar, confectioner’s sugar, and salt together until light and fluffy.
Blend in the egg, vanilla extract, and milk until well incorporated. The mixture will look slightly lumpy.
Add the flour mixture, beating just until combined and the dough starts to come together. Do not overmix.
With your hands, form the dough lightly into a uniform disc. Wrap in plastic wrap and refrigerate until firm, about one hour.
Cut and work with one half of the dough at a time, keeping the other half wrapped and refrigerated. Take out the dough and let it sit for about 5-10 minutes at room temperature. On a lightly floured board, roll out the dough with a rolling pin to approximately 1/4 inch thick. Use more flour if the dough gets sticky.
Cut as desired, and carefully transfer to a lined baking sheet. Repeat with the other half.
Once the cutouts are all transferred to baking sheets, place them in the refrigerator. Chill for 15 minutes.
Remove from the refrigerator and immediately bake until set, approximately 8-10 minutes. They will still be pale white and soft to the touch. Remove from oven. Let the cookies stand for 1 minute, then transfer them to write racks to cool completely.