In the bowl of a stand mixer or a large bowl with a hand mixer, cream together the butter, granulated sugar, and powdered sugar until light and fluffy, about 2 minutes.
1 cup unsalted butter, ½ cup granulated sugar, ½ cup powdered sugar
Add the egg and vanilla extract and mix until fully combined, scraping down the sides as needed.
1 large egg, 1 teaspoon vanilla extract
In a separate bowl, whisk together the flour, baking soda, cream of tartar, and salt. Gradually add the dry ingredients to the wet ingredients and mix just until combined.
2½ cups all-purpose flour, ½ teaspoon baking soda, ½ teaspoon cream of tartar, ½ teaspoon salt
Divide the dough into two equal portions. Place each portion between two sheets of parchment paper and roll into a rectangle about ¼-inch thick. Flip the parchment once or twice while rolling to keep the thickness even.
Transfer the rolled dough (still between parchment) to a baking sheet and freeze for 15 minutes, or until firm but not rock hard.
Preheat the oven to 350°F and line a baking sheet with parchment paper.
Lightly flour a clean work surface. Peel the parchment from the chilled dough and place it on the surface. Lightly dust the top with flour and cut into shapes using cookie cutters.
Gather scraps, re-roll to ¼-inch thick, and cut again. If the dough becomes soft, return it to the freezer for 10–15 minutes.
Transfer cookies to the prepared baking sheet, spacing about 1 inch apart. Bake for 8–11 minutes, just until the edges look set and barely start to turn golden.
Let cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely before decorating.