Adjust oven rack to the middle position and preheat oven to 400 degrees F. Whisk together the flour, baking soda, cream of tartar and 1/4 teaspoon salt in a medium bowl.
Add the butter, 1 cup of the granulated sugar, brown sugar and egg to the bowl of a stand mixer fitted with a paddle attachment (or a large bowl if using a hand mixer). Beat on medium speed just to combine, about 30 seconds. Add the flour mixture in 2 parts and beat on low until just combined. Let the dough rest at room temperature for 30 minutes.
Whisk together the remaining 1/4 cup granulated sugar and cinnamon in a small bowl.
Form the dough into 1 1/4-inch balls (about 1 tablespoon dough per cookie). Roll each ball in the cinnamon-sugar and coat well; arrange on two baking sheets, 2 inches apart. Do not flatten the balls, as they will spread a lot in the oven.
Bake one sheet of cookies at a time until the tops look cracked and the center still feels soft, about 9 minutes. Let the cookies cool in the pan for 3 minutes; they will continue cooking during this time. Once they've firmed up a little, transfer them to a rack to continue cooling. Store the cooled cookies in an airtight container for up to 3 days.
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