Beat butter and white sugar in a large bowl until creamy. Beat in egg, milk, and vanilla until smooth.

Whisk flour, baking powder, and salt in a separate bowl. Gradually stir dry ingredients into wet ingredients to make a smooth dough. Chill dough in refrigerator for 2 to 3 hours.

Preheat the oven to 400°F (200°C). Dust a pastry cloth or kitchen towel generously with flour.

Divide dough into thirds and roll each third out 1/8-inch-thick on the prepared pastry cloth. Cut shapes out of the rolled dough with cookie cutters. Place cookies onto ungreased baking sheets.

Bake in the preheated oven until cookies are barely browned, 5 to 8 minutes. Let cookies cool on baking sheets for 5 minutes before removing to finish cooling on a wire rack.

While cookies are cooling, prepare frosting: Beat 3 cups powdered sugar, butter, 9 teaspoons evaporated milk, and vanilla in a bowl until smooth. Gradually stir in remaining 6 cups powdered sugar into mixture until combined.

Continue to beat on high speed until frosting is fluffy; stir in additional evaporated milk or confectioners' sugar if needed to reach desired consistency.