Beat the butter and sugar on medium until light and fluffy, about 3 minutes. Add the egg and vanilla, and beat until combined.

Whisk flour, baking powder, and salt in a separate bowl. Gradually mix into the wet ingredients on low just until a soft dough forms. Flatten into a disc, wrap, and chill at least 1 hour.

Preheat oven to 350°F (175°C). Roll dough to ¼-inch thickness, cut shapes, and bake 8 to 10 minutes until edges are barely golden. Cool on sheet 5 minutes, then transfer to a wire rack.

Once completely cool, decorate with icing and sprinkles. Let icing set before stacking.