In a medium saucepan, melt the butter over medium heat. The butter may start to boil, just keep stirring the butter, (it may get foamy) until it turns a deep amber color. (This may take 5 to 10 minutes.)
Remove it from the heat and let it cool for at least 20 minutes.
While the butter cools, preheat the oven to 350 degrees.
In a large bowl, whisk together the flour, baking soda, and sea salt. Then set it aside.
Add the brown sugar, granulated sugar, eggs, and vanilla to the cooled butter and stir to combine. Pour this mixture into the bowl with the flour mixture. Stir it until the dough comes together and the ingredients are incorporated. Stir in the chocolate chunks.
Using a medium spoon (about 2 tablespoon size), scoop the dough, form it into a slight ball and drop it about 2 inches apart on the baking sheet. Bake for 10 to 12 minutes, until the edges are slightly golden.
Allow the cookies to cool about 5 minutes, then transfer them to a wire rack, or some aluminum foil to cool completely. If desired (and I recommend it) sprinkle the top with a little sea salt. Serve immediately.
These cookies are best eaten right after they're baked. The tend to lose that great brown butter flavor if they sit a day or two.