Step 1In the bowl of a stand mixer, or in a large bowl using an electric mixer, cream together the butter and sugar. Beat in the egg until combined.
Step 2Stir in the vanilla, then the baking powder. Slowly stir in the flour until combined. Wrap the dough tightly in plastic wrap and refrigerate until well-chilled, at least 2 hours.
Step 3Heat the oven to 425 degrees. Roll out the dough on a lightly floured surface about one-eighth inch thick. Cut the dough into desired shapes. Place the cookies approximately 1 inch apart on an ungreased baking sheet.
Step 4Bake the cookies in the center of the oven until set, about 5 to 7 minutes (they should not color and may only be faintly golden around the edges). Remove and cool the cookies on a rack. Frost the cookies (below) when cooled, or cover tightly and freeze until ready to frost, up to 2 months; thaw before frosting.
Step 1In a medium bowl, whisk together the powdered sugar and milk to form a glaze. Frost the cookies with a thin coating of the glaze, then decorate as desired.