Place a rack in the center of the oven then preheat to 400 degrees. Coat a baking sheet with cooking spray or line with parchment paper then set aside. In a medium-sized mixing bowl, sift together flour, cream of tartar, baking soda and salt.
In a separate mixing bowl or the bowl of a stand mixer, cream together butter and sugar until light and fluffy. Add eggs one at a time, incorporating the first egg completely before adding the next. Scrape down the sides of the bowl and beat in the vanilla.
Fold in dry ingredients until completely and thoroughly combined. Cover bowl with plastic wrap then chill in the fridge for 1 hour. After the dough has chilled, divide into walnut-sized balls—you should get about 2 dozen.
Mix the cinnamon and sugar in a small bowl then roll each of the dough balls in the mixture to coat evenly. Place on the baking sheet approximately 2" apart. Press each ball flat with the bottom of a drinking glass or coffee cup until ½" thick. If it begins to stick, just dip the bottom of the glass in the cinnamon sugar.
Bake 8–10 minutes until they have a crackled appearance and are lightly browned around the edges. Remove from oven, transfer to a cooling rack and enjoy!