Heat 1 stick (8 Tablespoons or ½ cup) of the butter in a small skillet over medium heat, stirring occasionally. The butter will melt and bubble. When the butter turns golden brown and becomes fragrant, remove from heat and pour into a heat proof bowl or measuring cup to cool. Place in the fridge or freezer to cool more quickly.
Mix together the flour, cream of tartar, salt, and baking soda in a medium bowl with a whisk. Set aside.
Place the remaining stick of butter into the bowl of a stand mixer or large bowl. Add the cooled browned butter, granulated sugar and light brown sugar. Mix on medium high speed until the mixture is uniform and fluffy, about 4-5 minutes. Add the eggs and vanilla and mix on medium high speed again until light and fluffy, about 3 minutes.
Place the dough in the fridge and chill for 1 hour. Alternatively, place in the freezer for 15-20 minutes if you're in a rush.
Move oven racks to top middle and bottom middle positions and heat oven to 350°F. Prepare 2 baking sheets by lining with parchment paper or silicone baking mats. Stir together the sugar and cinnamon in a small bowl for the cinnamon sugar coating.
Portion the dough into 1 ½ to 2 Tablespoon portions and roll into smooth balls in the palms of your hands. Roll the balls in the cinnamon sugar mixture until coated completely. Place the balls about 2 inches apart on the baking sheets and flatten slightly with the palm of your hand.
Bake until the edges of the cookies have set, about 9-12 minutes, switching and rotating the baking sheets in the oven halfway through the baking time. Let the cookies cool on the baking sheets for 10 minutes, then transfer to a wire cooling rack to cool completely. Store extras in an airtight container at room temperature for 2 days, or freeze.