Cream the butter and sugar together until pale and moving towards moussiness, then beat in the egg and vanilla. In another bowl, combine the flour, baking powder, and salt. Add the dry ingredients to the butter and eggs, and mix gently but surely. If you think the finished mixture is too sticky to be rolled out, add more flour, but do so sparingly as too much will make the dough tough. Form into a fat disk, wrap in plastic wrap, and let rest in the refrigerator for at least 1 hour.
Sprinkle a suitable surface with flour, place disk of dough on it, and sprinkle a little more flour on top of that. Then roll it out to a thickness of about 1/4-inch. Cut into shapes, dipping the cutter into flour as you go, and place the cookies a little apart on 2 parchment or silpat lined baking sheets.
Bake for 8 to 12 minutes; obviously it depends on the shape you're using and whether they are on the upper or lower shelf, though you can swap them around after about 5 minutes. When they're ready expect them to be tinged a pronounced gold around the edges; they'll be softish still in the middle, but set while they cool.
Remove the cookies with a flat, preferably flexible, spatula to a wire rack. When they are fully cooled, you can get on with the icing. Put a couple of tablespoons of not-quite-boiling water into a large bowl, add the sieved confectioners sugar and mix together, adding more water as you need to form a thick paste. Color,as desired. I think pastes are much better than liquid, not just because the range of colors is better but because they don't dilute the icing as they tint. Ice cooled cookies, as desired.