Heat a large, heavy-bottom skillet over medium heat. Add all of the butter at once and allow to melt, stirring continuously, until it begins to foam and turn brown in color, about 5 minutes. Continue to stir over medium heat until the foam has subsided. Remove the pan from heat and continue to stir until the butter is a medium golden brown and has a nutty aroma. Allow to cool for 10 minutes.

Place unwrapped chocolate kisses in the freezer.

Combine flour, baking soda, baking powder, and salt in a medium bowl. Set aside.

Combine browned butter, brown sugar, and white sugar in a large mixing bowl. Beat with an electric mixer on medium speed until well combined. Add eggs and vanilla extract; mix well. Slowly incorporate flour mixture into butter mixture, mixing until well combined; do not overmix. Allow dough to chill in the refrigerator for 40 to 60 minutes.

Preheat the oven to 350 degrees F (175 degrees C). Line baking sheets with parchment paper.

Drop chilled dough by tablespoonfuls onto the prepared baking sheets.

Bake in the preheated oven until the edges are light golden brown, 10 to 13 minutes.

Allow cookies to cool on baking sheet for exactly 4 minutes. Gently press a caramel-chocolate kiss into the center of each cookie; press slightly into cookie and not just placed on top. Sprinkle a scant pinch of sea salt over the center of the cookies and allow to cool completely on a cooling rack. Refrigerate if necessary to re-harden caramel kisses.