Preheat the oven to 375°F and line a baking sheet with parchment paper or a silpat.
Whisk together flour, baking powder, and salt in a medium bowl; set aside.
Beat butter and sugar in a stand mixer until light and fluffy.
Add the egg and almond extract to the butter mixture and mix until combined.
Gradually add the flour mixture, half a cup at a time, mixing until fully incorporated.
Roll tablespoon-sized portions of dough into small circles and press into disks with your hands or the bottom of a cup.
Place the shaped cookies on the prepared baking sheet.
Bake for 8 minutes; cookies will not appear browned but are fully cooked.
Let cookies rest on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
Prepare icing by whisking powdered sugar, 1 tablespoon milk, and almond extract in a small bowl until smooth; add more milk if needed to thin.
Spread the icing onto the cooled cookies using a spoon and allow to set for 10 minutes until hardened.
Store iced cookies in the fridge and bring to room temperature before serving.