Beat the butter and powdered sugar on medium-low for 60 to 90 seconds, just until smooth. Do not beat until light and fluffy — the air you add will puff the design flat. The mixture should look dense and paste-like.
Mix in the vanilla on low, about 15 seconds.
Whisk the flour, cornstarch, and salt together in a separate bowl, then add all at once. Mix on low only until no dry streaks remain. The dough will look crumbly, then come together suddenly.
Roll the dough directly on a sheet of parchment to a scant 3/8 inch thick. Leave it at room temperature — chilled dough resists the stamp and the imprint comes out shallow.
Dip the stamp in flour and tap off the excess. Press straight down with firm even pressure, then lift straight up. Re-flour before every press. Work across the whole sheet, leaving just enough room between imprints for the cutter.
Center a fluted cutter slightly larger than the design over each imprint and cut, but leave the cookies in place. Peel away the excess dough from around them, then slide the whole sheet of parchment onto a baking tray. The dough is too soft to lift cookie by cookie.
Freeze on the tray for 30 minutes, until hard to the touch. This is the step that keeps the design sharp — frozen dough sets before the butter can melt and slump.
Bake at 350°F for 11 to 12 minutes, straight from the freezer. They're done when the bottoms are pale gold and the tops still look matte and blond. Don't wait for the tops to brown.
Rest on the pan 5 minutes, then move to a rack. They firm up as they cool.