Place 1 stick (8 tablespoons) of the butter in a small saucepan over medium-high heat. Cook, stirring almost constantly, for about 3 minutes or until the butter browns and gives off a slightly nutty aroma.
Remove the pan from heat, add the remaining 1 stick of butter, and whisk until the mixture is completely smooth. Transfer the butter to a bowl and cool it until it returns to room temperature (soft but not liquid). You can speed cooling by briefly chilling the bowl in the refrigerator or freezer; total time to reach room temperature is about 30 minutes. You’ll know it’s ready when pressing a finger makes a slight indentation.
While the butter cools, unwrap the 20 caramels and roll each into a small ball. Place the caramel balls in the freezer until firm.
Preheat the oven to 375°F. Line two large baking sheets with parchment paper and set aside.
In a medium bowl, whisk together the 2 1/2 cups all-purpose flour, 1 teaspoon baking soda, and 1/4 teaspoon salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl with a handheld mixer, combine the cooled browned butter, 1 cup + 2 tablespoons dark brown sugar, and 1/2 cup granulated sugar, and 2 teaspoons vanilla extract. Beat on medium-high speed until the mixture is light and fluffy, about 2 minutes.
Reduce the mixer speed to medium and add the 2 large eggs one at a time, beating well after each addition.
Let the mixture rest for 5 minutes.
Gently fold the flour mixture into the butter mixture with a rubber spatula, stirring only until the flour disappears. Fold in the 2 1/2 cups semi-sweet chocolate chunks until evenly distributed.
Using a 2-tablespoon measure, scoop a 2-tablespoon portion of dough and roll it into a ball. Place the dough ball on the prepared baking sheet, flatten slightly, and press a frozen caramel ball into the center. Top with a second 2-tablespoon portion of dough, cover the caramel, and pinch the edges together to seal completely. Place the sealed cookie seam-side down on the sheet. Repeat with the remaining dough and caramels, spacing the cookies about 2 inches apart.
If your oven can only handle one sheet at a time, bake the cookies one sheet at a time; keep the unbaked sheet in the refrigerator for a few minutes between batches if the dough becomes soft.
Bake in the preheated oven for 11–14 minutes, or until the edges are golden but the centers still look slightly soft.
Remove the baking sheet from the oven and let the cookies cool on the sheet for 5 minutes. Transfer the cookies to a wire rack to cool completely.