Place the butter in a light-colored saucepan over medium heat. Stir frequently as it melts, foams, and begins to turn golden with brown specks on the bottom. Once it smells nutty and is amber in color, about 5 to 7 minutes, immediately pour it into a large heatproof bowl. Let it cool for 15 minutes.
While the butter cools, whisk together gluten free all-purpose flour blend, baking soda, cornstarch, and salt in a medium bowl. Set aside.
Add the dark brown sugar and granulated sugar to the cooled brown butter and whisk vigorously until smooth and combined. Whisk in the egg, egg yolk, and vanilla extract until the mixture is thick and glossy.
Add the flour mixture and stir with a spatula until a thick dough forms with no dry streaks remaining. Cover and refrigerate the dough for at least 1 hour.
Preheat your oven to 350 degrees F and line two baking sheets with parchment paper. Scoop about 2 tablespoons of dough, flatten it slightly, press one caramel candy into the center, and wrap the dough around it to seal completely. Roll into a ball and place seam-side down on the baking sheet, spacing 3 inches apart.
Bake for 11 to 13 minutes until the edges are golden but the centers are still puffy and soft. Immediately sprinkle flaky sea salt over each cookie.
Let the cookies cool on the baking sheet for 10 minutes before transferring to a wire rack. The caramel center will be perfectly gooey when served slightly warm.