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