Preheat the oven to 350 F/180 C. Line a square 8x8 baking pan** (see notes below) with parchment paper and set aside.
In a stainless steel pan, brown the butter over medium heat for 10-12 minutes. While it cooks, the butter will foam, pop, and crackle. This is normal! Stand by and stir it occasionally. When the bottom of the pan has brown bits of butter solid and it starts to smell nutty, scrape all of it into a separate bowl.
Add the chocolate chips to the bowl of browned butter and stir to melt completely. Set aside.
In a bowl, whisk together the flour, cocoa powder, espresso, and salt. Set aside
In a mixing bowl, whisk the eggs, sugar, and vanilla together on medium speed for 1-2 minutes until fluffy.
With the mixer running on low, stream the melted butter and chocolate into the batter. Mix until just combined.
Gently fold in the dry ingredients with a rubber spatula until just combined.
Pour the batter into the prepared baking pan. Bake for 28-35 minutes, or until a toothpick inserted into the center comes out covered in moist crumbs.All ovens are different, and your brownies may bake quicker or take longer than the indicated baking time. If your pan has rounded edges or is anything other than METAL, it will take longer to bake, closer to 45-50 minutes. Let the pan cool completely on a wire rack then cut into 16 squares with a hot sharp knife, wiping it clean between cuts.