Make the snickerdoodle cookies: Line 2 baking sheets with parchment paper. Whisk together the flour, cream of tartar and kosher salt in a medium bowl until evenly combined. Whisk 1 cup of the sugar, the butter, vanilla and egg together in a large bowl until well combined. Pour in the flour mixture and stir with a spoon until the dough just combines.
Stir together the remaining 1/4 cup sugar and the cinnamon in a small bowl. Using a 1/2-ounce ice cream scoop or a tablespoon, portion and roll the dough into 30 (1-inch) balls. Roll each ball in the cinnamon-sugar to coat, and then transfer to the prepared baking sheets, spacing the balls 2 inches apart. Using the end of a wooden spoon or your index finger, press each dough ball in the center to create a deep divot. Refrigerate the dough balls on the baking sheets for 30 minutes.
Preheat the oven to 400 degrees F. Bake the cookies, rotating the baking sheets front to back and top to bottom halfway through, until light golden brown on the bottom, about 12 minutes. Let the cookies cool for 1 minute on the baking sheets, and then transfer them to a wire rack to cool completely.
Make the salted caramel filling: Pour the sugar in a small saucepan and place over medium heat. Cook the sugar, stirring occasionally, until it turns liquid and deep amber brown. Remove the pan from the heat and stir in the butter until smooth. Add the cream and kosher salt and stir until incorporated. Fill the divot in each cookie with the hot caramel and sprinkle with sea salt. Let the caramel cool for 30 minutes to set before serving.
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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