Preheat your oven to 350-F degrees.
Make the brown butter by melting the butter in a saucepan over medium heat. Stir the butter until it foams up, then take it off the heat and continue whisking until the foam subsides. You should see that the butter solids have turned golden brown, if not return the pan to the heat and stir until the butter has brown flecks in it and has a nutty aroma. Remove from the heat and cool to room temperature.
Combine the flour, cream of tartar, baking soda, cinnamon, and salt in a bowl.
Mix the brown butter, brown sugar, and granulated sugar in the bowl of a stand mixer until well combined and smooth.
Add the egg and egg yolk, and vanilla and mix until just combined.
Add the dry ingredients in 2 batches, mixing well in between.
Turn the dough onto a sheet of plastic wrap and wrap it tightly. Refrigerate the dough for about an hour.
Scoop the dough into 2 tablespoon balls (a cookie scoop works fabulously for this). Press your thumb into each dough ball to create a depression, then add a caramel to the hole. Work the cookie dough over the caramel, sealing the caramel inside the middle of the cookie ball.
Roll each dough ball in the cinnamon sugar then bake them 8 at a time on a parchment paper-lined baking sheet for about 10-12 minutes in a preheated 350-F oven or until the cookies are cracked on top.
Sprinkle the cookies with the flake salt as soon as they come out of the oven. Let them cool on the baking sheet before removing them. Enjoy!