Beat butter and sugar 2 minutes until smooth and slightly fluffy. Beat in egg and vanilla, scraping bowl.
Whisk flour, baking powder, and salt; mix into butter on low just until dough forms. Flatten into two discs; wrap and refrigerate 1 hour, additional to listed active time.
Heat oven to 350°F; parchment-line two sheets. Roll discs between parchment to 1/4 inch thick. Use simple 2 1/2-inch cutters, rerolling scraps once, for about 24 cookies.
Bake one sheet at a time 10 to 11 minutes until edges set and bottoms pale gold. Cool on sheet 5 minutes; transfer to rack.
Whisk sugar and water into thick spreadable icing. Thinly spread on fully cooled cookies; dry 1 hour before stacking.