Preheat the oven to 325°. Line a 9x13 pan with parchment paper or spray with oil.
In medium sized pan add the butter and cook over medium heat, whisking, butter will foam up and then calm down, keep whisking until you see the butter start to turn a golden brown, it'll take 2-3 minutes. It'll give off a nutty aroma. Remove from heat and allow to cool slightly.
Meanwhile, place your dark brown sugar and regular sugar in the bowl of a mixer. Combine sugars with vanilla, and slowly pour in browned butter. Mix to combine. Add eggs and mix to combine. Add flour and salt and mix until just combined, careful not to over mix. Scrape down sides.
Spread evenly into the bottom of your prepared 9x13 pan, The dough will be thick, like peanut butter. Bake for 18-20 minutes or until the edges are lightly brown and just starting to pull from sides. Remove from oven and allow to cool on a wire rack.
While cooling make frosting.Melt butter and packed light brown sugar, bring to simmer. Add ¼ cup milk and return to simmer whisking constantly. Remove from heat and add powdered sugar, whisking until incorporated and lumps are gone. If needed, use hand mixer for smoothness. May frost warm or cooled.
Frost cookies and sprinkle with sea salt, Fleur de Sel is wonderful, but not necessary. Cut into 2x2 squares Cover any leftovers tightly, if there are any! May be frozen.