Line a large baking sheet with nonstick foil or parchment paper. This will not be the tray you bake on, but the tray that holds the dough while it chills.
In a large bowl, thoroughly mix together the flour, baking powder and salt. Set aside.
In a mixing bowl, beat butter until creamy. Add ground sugar and beat on high for two minutes or until mixture is creamy and smooth. Beat in the vanilla, scraping sides of the bowl. Add the egg and beat until mixed.
Gradually add flour mixture and stir to form a very stiff, dry dough. If you are using a handheld mixer, put the mixer down and use your hands to form the dough. If you are using a stand mixer with a paddle, you can let the paddle do the work. Dough might be more like crumbs, but they should hold together when you press them down.
Empty the flour mixture out onto the lined tray and pat it down into a slab that's about ½ inch thick. You won't be able to cut it at this point because it's too crumbly. Using your hands, mash it inward and make it a tight slab. Cover with a sheet of nonstick foil or parchment and chill for at least an hour (preferably longer) or until the slab is very firm and holds together.
You will not need any extra flour when cutting shapes. Using a 2 inch to 2 ½ cookie cutter, punch shapes out of the dough slab and carefully (they are delicate) arrange them on baking sheets.
Bake in a 350 degree F. oven for about 12 to 15 minutes or until edges are very lightly browned. Let cool on baking sheets for about 5 minutes, then carefully transfer to a cooling rack. They will become sturdier as they cool.
To make the frosting, beat the butter until creamy. Beat in the salt (omit if using salted butter), then add the sugar gradually and beat until smooth. Beat in vanilla. If necessary, add a little whole milk to get the consistency you want. Decorate with this frosting or if transporting, use a royal frosting. This one crusts over, but will not get hard.