Preheat oven to 350°F. Line a sheet pan with parchment paper or a silicone baking mat.
In the bowl of a stand mixer fitted with a paddle attachment, or in a bowl with a hand-held mixer, beat the butter, sugars and vanilla together until smooth and creamy, about 3-5 minutes. Add the egg andand mix until just combined.
In a separate bowl, combine the flour, baking soda, salt, and cream of tartar. Add to the mixer and mix on low until combined. Mixture will be crumbly at first, but will eventually come together into a thick dough.
Dump onto a clean surface and massage the dough a bit until it comes together. Shape into a disk.
With a rolling pin, roll dough out to about ¼ inch thick (up to ½ inch for super thick cookies), adding more flour to your work surface and to the dough as needed to keep it from sticking.
Cut dough out with cookie cutters into desired shapes and transfer to a sheet pan lined with parchment paper, leaving several inches of space between each cookie.
If desired, freeze cookies on the pan for 15 minutes (this is extra insurance that your cookies will no spread and lose their shape while baking, but is not required).
Bake until cookies are puffed and the tops appear completely dry, about 8-12 minutes. Baking time will vary based on size of the cookies (see Notes). Let cool on the pan 5 minutes, then transfer to a sheet pan to cool completely.