Preheat the oven to 350°F (180°C). Grease and line an 8×8-inch baking pan with parchment paper, leaving some overhang for easy removal.
Place the dark chocolate pieces into a heatproof mixing bowl.
Add the butter to a saucepan over medium heat. Stir continuously as it melts and begins to foam.
Continue cooking until the butter turns golden brown and smells nutty and caramel-like. Remove from heat immediately.
Pour the hot brown butter over the dark chocolate. Stir until the chocolate melts completely and the mixture is smooth. Set aside.
In a large bowl, whisk together the eggs, brown sugar, and granulated sugar for 3–4 minutes until pale and slightly fluffy.
Add the vanilla extract and mix well.
Pour the brown butter chocolate mixture into the egg mixture and whisk until fully combined.
Add the flour and salt. Gently fold with a spatula until just combined. Do not overmix.
Fold in the chopped milk chocolate and chocolate chips.
Pour the batter into the prepared pan and smooth the top evenly.
Bake on the middle oven rack for 35–40 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs.
Allow the brownies to cool completely in the pan before removing and slicing.
Serve and enjoy with a cold glass of milk.