Start by melting the unsalted butter in a medium saucepan over medium heat. Stir frequently until it foams and turns golden brown with a nutty aroma, about 5-7 minutes. Remove from heat and cool slightly.
In a large mixing bowl, whisk together the brown sugar and slightly cooled browned butter until smooth. Add the lightly beaten eggs and vanilla extract, mixing until fully combined.
In another bowl, whisk the all-purpose flour, baking soda, and salt. Gradually add the dry mixture to the wet ingredients, stirring until just combined; a few lumps are okay.
Gently fold in the milk chocolate and dark chocolate chips using a spatula for even distribution.
Preheat the oven to 350°F (175°C). Line a 9x13 inch baking pan with parchment paper with some overhang. Lightly grease the parchment.
Spread the cookie bar batter evenly into the prepared pan and smooth the top. Bake for 20-25 minutes until edges are golden brown and a toothpick comes out with moist crumbs.
Sprinkle a generous amount of Maldon flaked salt on top of warm cookie bars to enhance flavor.
Allow bars to cool in the pan for about 15 minutes, then lift out using parchment overhang and transfer to a wire rack to cool completely. Slice into squares or rectangles and enjoy.