Preheat your oven to 350°F and line an 8x8 inch baking pan with parchment paper.
In a medium saucepan, melt the butter over medium heat. Continue cooking until it foams and turns a golden brown color, about 5-8 minutes. Remove from heat and let cool slightly.
In a large bowl, whisk together the granulated sugar, brown sugar, and eggs until smooth. Stir in the vanilla extract.
Gradually add the cooled brown butter to the sugar and egg mixture, mixing until well combined.
Sift together the flour, cocoa powder, and salt. Gradually fold the dry ingredients into the wet ingredients until just combined. Avoid overmixing.
If using, fold in the chocolate chips.
Pour the batter into the prepared pan and smooth the top with a spatula.
Bake in the preheated oven for about 35-40 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs.
Allow the brownies to cool in the pan before slicing into squares.