Melt 10 tablespoons butter in a 10-inch skillet over medium-high heat. Continue to cook and stir constantly until butter is dark golden brown, 1 to 3 minutes. Transfer browned butter to a large heatproof bowl. Add remaining 4 tablespoons butter (chopped) and stir until completely melted; set aside to cool.
Meanwhile, preheat oven to 350 F degrees. Line 2 baking sheets with parchment paper or non-stick mats. Set aside.
Whisk flour, baking soda, and baking powder together in medium bowl; set aside.
Add brown sugar and salt to bowl with cooled butter and whisk until combined. Add egg and yolk and vanilla and mix until combined. Add flour mixture and mix just until completely combined.
Scoop 1 tablespoon of cookie dough and place a caramel on top. Add another tablespoon of dough on top of caramel, pinch edges of dough around caramel and roll into a ball. Repeat until all remaining dough is gone.
Place cookies on prepared baking sheets 3 inches apart and sprinkle with sea salt. Bake at 350 degrees F 11-13 minutes until barely golden (they will seem underbaked). Let cool for 5 minutes before transferring to cooling rack.
Store leftovers in an airtight container for up to 7 days. I like to microwave cooled cookies for 10 seconds or so for softer caramel.