In a medium bowl, whisk together the flour, cornstarch, cream of tartar, cinnamon, and baking soda; set aside. In a stand mixer fitted with the paddle attachment, beat the butter and both sugars on medium speed until light and fluffy, 3 to 4 minutes.
Next, with the machine running on low speed, add the eggs one at a time until incorporated, stopping and scraping down the sides of the bowl after each addition. Then add the vanilla and mix on low speed until fully incorporated.
With the machine running on low speed, add the flour mixture and mix just until you no longer see any bits of flour. Transfer the cookie dough to a medium bowl, cover it tightly, and chill in the refrigerator for at least 30 minutes.
If chilling for 3+ hours, make sure you let the cookie dough sit at room temperature for at least 30 minutes before rolling it into balls; the cookie dough will be very stiff after being in the fridge that long.
Meanwhile, in a small bowl, mix ⅓ cup of granulated sugar and 2 teaspoons of ground cinnamon; set aside.
Preheat the oven to 350 °F. Position racks in the upper and lower thirds of the oven. Line two baking sheets with parchment paper; set aside. (If you only have 1 baking sheet, let it cool completely between batches). Using a 1.5-inch ice cream or portion scoop, scoop the dough into rounded tablespoon-sized rounds, scraping each against the bowl as you scoop.
Roll each mound in your hands to form a ball. The dough will be very soft, so handle it carefully and work quickly. Drop each ball into the cinnamon and sugar mixture and roll around to coat thoroughly. Place on the prepared baking sheet, about 2 to 3 inches apart. Repeat with the remaining dough balls.
Bake one sheet at a time until the snickerdoodles cookies have puffed up and the tops begin to crackle, 10 to 12 minutes; do not overbake. The longer you bake the cookies, the crispier they’ll become. Remove from the oven and 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.