Sift the flour, baking powder, and salt together into a medium bowl and set aside.
Place butter and sugar in the bowl of a stand mixer fitted with the paddle attachment. Start the mixer on low, then gradually increase the speed to medium and continue to beat, scraping down the sides of the bowl as needed, until light and fluffy, 3 to 5 minutes, depending on the temperature of the butter.
Slow the mixer to low and add the egg, followed by the milk, then return to medium speed and beat until smooth, about 30 seconds. Then, slow the mixer yet again, gradually add the flour, and mix until dough comes together, about 30 more seconds.
Remove the dough from the bowl and split into two equal portions. Form these into balls, wrap each piece in plastic wrap, and refrigerate for 1 hour.
Preheat oven to 375 degrees F with the oven racks in the upper middle and lower middle positions. Line two half sheet pans with parchment paper.
Use a small strainer or sifter to sprinkle your rolling surface with confectioners' sugar. Remove 1 dough from the refrigerator, place on the sugar-dusted counter, and dust with a little additional sugar. Working quickly (so the butter doesn't have time to melt), roll the dough to a quarter-inch thickness. (I use pastry guides for this.)
Cut dough into desired shapes and place at least 1 inch apart on the prepared sheet pans. Re-roll the scraps one time and cut into additional cookies. Discard any remaining dough; do not re-roll a second time. (Okay, I admit I eat it raw, but that's me.) Bake both pans of cookies, rotating halfway through baking time, until the cookies are just brown around the edges, about 10 minutes. Remove from oven and cool on the pans for 2 minutes, then transfer to a wire rack to cool completely.
While the first batch is baking, roll and cut the second half of the dough. Bake and cool the second batch, then decorate as desired…or don't. Frankly, they’re plenty good with no icing at all.