In a medium bowl, whisk together the flour, cream of tartar, baking soda and salt.
In the bowl of stand mixer fitted with the paddle attachment, cream together the butter and 1 1/2 cups of sugar until light and fluffy, scraping down the sides as needed. Add the eggs, one at a time, beating well after each addition. Add the flour mixture, beating just until combined.
Cover the bowl with plastic wrap and refrigerate the dough for a minimum of 2 hours and up to 3 days.
When ready to bake, preheat the oven to 350°F and line two baking sheets with parchment paper or Silpats.
In a small bowl, combine the remaining 1/4 cup sugar with the cinnamon.
Using a spoon, scoop out about 2 tablespoons of cookie dough and flatten it slightly. Roll a soft caramel into a ball, and then place it in the center of the dough and cover the caramel completely with the dough. Roll the cookie into a ball then lightly roll it in the cinnamon-sugar mixture on all sides. Place the cookie on the baking sheet then repeat the filling process with the remaining dough, spacing the cookies at least 2 inches apart. Sprinkle the tops of each cookie with a small pinch of flake salt.
Bake the cookies for 12 to 15 minutes until they are slightly underbaked. Remove the cookies from the oven and let them cool on the baking sheets for 10 minutes. (See Kelly's Notes.) Transfer the cookies to a wire rack to cool completely.