In a mixing bowl, combine the powdered sugar, and softened butter. Cream together with an electric hand mixer, until smooth. Add in the egg, and vanilla extract. Cream together.
Add in the flour, baking soda, and cream of tartar. Starting on low, gently increase the speed of the hand mixer, until all the ingredients are mixed and the dough is smooth.
Lay a piece of plastic wrap flat on the counter, transfer the cookie dough to the plastic wrap. Shape the cookie dough into a disk, then wrap the plastic wrap tightly around the cookie dough.
Refrigerate for 3 hours.
Preheat the oven to 375°.
Take the dough out of the refrigerator. Let the dough sit just long enough for it to soften a bit, so it can be rolled out. Place the dough on the floured surface, then lightly flour the top of the dough.
Using a rolling pin, roll the dough out so it's a 1/4" thick. Using the cookie cutters, cut the cookie dough into shapes. Take the cookie dough scraps, shape them back into a ball. Re-flour the counter and the top of the cookie dough and roll it out again. Then cut into more shapes. Repeat the process, until all the dough has been cut into shapes.
Place a few cookies on a cookie sheet, with a 1/2" between each cookie. Bake for 11 minutes, or until the edges are golden brown.
Remove the oven and let cool for 10 minutes, transfer to a cooling rack. Let cool completely before icing.