Arrange oven rack in center of oven and preheat to 400°F (200°C). (If you can’t fit two cookie sheets on one rack, arrange in upper and lower thirds and plan to rotate during baking.) Line 2 cookie sheets with parchment.

Cream butter and sugar until light and fluffy with a hand mixer. Add in this order, mixing fully between each addition: the egg, vanilla, salt, and baking powder; beat until fluffy again. Add flour ½ cup at a time, beating to just incorporate each addition, and adding 1 to 2 tablespoons of cold water before last addition of flour to moisten dough. Dough should look like a bowl of pebbles after last addition of flour. Set mixer aside and knead dough in bowl until it just comes together, about 10 turns.

Roll (or pat or smush) dough between parchment to a ¼-inch (6mm) thick 9 ½-inch (24-cm) wide round. Pop into the freezer for 15 minutes until dough is hard. Remove top sheet of parchment and cut cookies as desired. Place cut dough on parchment-lined cookie sheets. Gather scraps and re-roll dough once (no need to freeze again).

Bake cookies until edges are just puffed and center has a slight crust/skin but hasn’t taken on any color around edges, 8 to 10 minutes. Remove from the oven and slide parchment onto cooling racks; let cookies cool completely before frosting with a nice schmear. Let frosting firm before serving. Store in an airtight container; cookies are best the next day.

Beat butter until smooth and fluffy, then gradually incorporate powdered sugar. Add milk and vanilla and beat until fluffy and lightened by a few shades and vanilla is evenly distributed.

Cover and keep cool—but don’t refrigerate—until ready to frost cookies, up to a few hours.