Preheat the oven to 350ºF and lightly flour your work surface.
Sift together the flour and baking powder. Set it aside.
In the bowl of a stand mixer fitted with the paddle attachment, cream together the sugar and butter.
Add the egg, and then add the vanilla extract.
Add the flour mixture to the butter mixture in thirds and beat it just until combined making sure to scrape down the sides of the bowl as needed.
Transfer the dough to the lightly floured surface and roll it until it's about 1/4-inch thick all over.
Using a pizza cutter or sharp knife, cut the dough into small squares measuring about 3/4-inch on all sides.
Transfer the squares to a parchment paper-lined baking sheet and bake the cookies for 6 to 8 minutes until they're pale golden brown.
Transfer the cookies to a cooling rack, and while the cookies cool, make the frosting.