Beat butter and sugar in a large bowl until light and fluffy, 2 to 3 minutes. Add egg and 2 tsp vanilla, beat until smooth.

Whisk flour, baking powder, and salt in a separate bowl. Gradually add to the butter mixture on low speed, mixing just until combined.

Press the dough into a disc, wrap in plastic, and refrigerate for at least 30 minutes.

Preheat oven to 350°F (175°C). Roll dough to 1/4 inch thick on a floured surface, cut shapes, and place on parchment-lined baking sheets.

Bake 8 to 10 minutes until edges are barely golden. Cool on the sheets 5 minutes, then transfer to a wire rack to cool completely.

Whisk powdered sugar, 2 tbsp milk, 1 tsp vanilla, and corn syrup until smooth. Thin with extra milk as needed, then spread over cooled cookies and let set about 30 minutes.