Line two baking sheets with parchment paper. Preheat the oven to 400 degrees F (204 C).
In a stand mixer using the paddle attachment, cream the sugar, butter, salt, and vanilla together until loosened up. Crack in the egg and beat on high until light and fluffy. Add the baking powder and GF flour and mix on low until barely combined, then turn the speed up to high and let it mix for a couple minutes.
On a clean surface dusted with some GF flour, take half of the cookie dough and place it on the surface. Lightly flour a rolling pin and roll out the dough to about ¼ inch thick. Using festive cookie cutters, cut out as many cookies as you can. Remove the edge pieces and transfer the cutout cookies, one by one to a prepared baking tray using a flat, metal serving spatula. Place each cookie about 1 inch apart from each other on the baking tray. Bake for 5-7 minutes on the middle rack, until the bottom of the cookies are ever so slightly golden brown. Remove from the oven and let them cool at least 10 minutes before removing from the warm pan.
While one tray of cookies is baking, combine the leftover cutout dough with the remaining half of dough. Re-dust the surface and rolling pin with GF flour and repeat step 3 until as much of the cookie dough is cut into cookies as possible (you can eat any remaining or bake it in a random shape if you're scared of raw eggs). Let the cookies cool completely while making the frosting.