Sift together the flour, baking powder, and salt. Set aside.
Place butter and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat until light in color, about 2 minutes. Add the egg and milk and beat to combine. With the mixer on low speed, gradually add the flour mixture, and beat until mixture pulls away from the sides of the bowl. Divide the dough in half, wrap each piece in parchment paper, and refrigerate for 2 hours.
Heat oven to 375ºF and place a half-sheet pan in the freezer to chill.
Use a small strainer or sifter to sprinkle your rolling surface with 1 tablespoon powdered sugar. Remove 1 wrapped pack of dough from refrigerator at a time, sprinkle a rolling pin with powdered sugar, and roll out dough to 1/4-inch thick. Move the dough around and check underneath frequently to make sure it is not sticking. If dough has warmed during rolling, place the chilled half-sheet pan on top for 10 minutes to chill.
Cut dough into desired shapes, place at least 1 inch apart on a parchment paper-lined half-sheet pan, and bake until cookies are just beginning to turn brown around the edges, 8 to 12 minutes, rotating the pan halfway through baking time. Leave on the pan for 2 minutes after removal from oven and then move to a wire rack to cool completely. Serve as is or ice as desired. Store in airtight container for up to 1 week.