Place 1 stick of butter (8 tablespoons) into a small saucepan over medium-high heat; cook for about 3 minutes - stirring almost constantly - or until the butter has browned and taken on a slightly nutty aroma. Remove from heat and stir in the remaining stick of butter; whisk until completely smooth. Place the bowl in the refrigerator or freezer and bring the butter back to room temperature, about 30 minutes. *You'll know the butter is at room temperature when you press a finger into the top and it makes a slight indentation.
Unwrap caramels and roll each one into a small ball; place in the freezer while the butter cools.
Once the butter is at room temperature...
Preheat oven the to 375 degrees (F). Line 2 large baking sheets with parchment paper; set aside. In a medium-sized bowl whisk together the flour, baking soda, an salt; set aside.
Combine the brown butter, sugars, and vanilla in the bowl of a stand mixer fitted with the paddle attachment or in a large bowl using a handheld electric mixer. Beat on medium-high speed until light and fluffy; about 2 minutes. Reduce the speed to medium and add in the eggs, one at a time, beating well after each addition. Let the mixture rest for 5 minutes. Gently fold in the flour mixture with a rubber spatula, stirring only until the flour disappears. Fold in the chocolate chunks and stir gently to combine.
Scoop out a two tablespoon round of dough, roll it into a ball, and place on prepared baking sheet. Gently press the frozen caramel into the center, then top with another 2 tablespoon sized round of dough, lightly pinching the edges closed to seal the caramel completely; repeat with all cookie dough.
Place baking sheets in preheated oven (preferably one at a time, keeping the other in the fridge for a few minutes) and bake for 11-14 minutes, or until golden at the edges but still slightly soft in the middle. Let cookies cool for 5 minutes on the sheet before transferring to a wire wrack to cool completely.