Preheat the oven to 350°F (175°C). Cream the butter and sugar with an electric mixer on medium speed until pale and fluffy, about 3 minutes.

Mix in the egg and vanilla on low, then add the flour, baking powder, and salt, mixing just until the dough comes together.

Shape the dough into a disk, wrap it, and chill at least 1 hour. Let it sit at room temperature 5 to 10 minutes, then roll to just under 1/2 inch thick on a lightly floured surface.

Dip cutters in flour, press straight down, and transfer shapes to a parchment-lined baking sheet 2 inches apart.

Bake 8 to 10 minutes for 1/4-inch thick cookies, until edges are barely golden. Cool on the sheet for 5 minutes, then move to a rack.

Whisk the powdered sugar, milk, and vanilla until smooth, tint with food coloring if you like, and spread or pipe onto cooled cookies. Add sprinkles while wet and allow icing to set, about 20 minutes.