Preheat the oven to 350 degrees F. Line two cookie sheets with parchment paper or silicone baking mats.
With a stand mixer or electric hand mixer, cream together the softened butter with one cup sugar, one egg and two teaspoons of vanilla extract.
In a separate bowl, whisk together the flour, cream of tartar, baking soda and salt. Add the flour mixture to the bowl with the butter mixture. Start on low speed and gradually increase the speed until a cohesive dough is formed. Scrap down the bowl as necessary with a rubber spatula while mixing. Refrigerate the dough for a little chilling for 20 minutes.
In a shallow small bowl, stir together the remaining 1/4 cup sugar with the cinnamon until well combined. Use a medium cookie scoop (1.5 tablespoon capacity) to form the cookie dough into 18 balls. Roll each ball in the cinnamon sugar mixture. Place 8 cookies on each lined baking sheet.
Bake at 350 for 11-12 minutes, rotating the baking pans front to back and top to bottom halfway through baking time. Allow the cookies to cool on the cookie sheets for 10 minutes before removing them to a wire rack to finish cooling.