Adjust the oven rack to a middle position and preheat the oven to 350°F (175°C). Line an 8"×8" pan with parchment paper (no need to grease) and set aside.
In a medium bowl, stir together eggs, sugars, and vanilla extract. Set aside and stir occasionally to help the sugar dissolve.
In a medium pot, melt the butter over medium-low heat. Once it's completely melted, simmer over medium heat, stirring occasionally with a heat-resistant silicone spatula; it will bubble and hiss. When it goes silent, remove from heat, stir once more to incorporate all the browned solids, and set aside.
In a small bowl, sift together the cocoa powder, flour, espresso powder, baking soda, cornstarch, and salt.
Slowly whisk the browned butter into the sugar/egg mixture, starting with just a few drops at a time, whisking all the time to heat the eggs slowly and not cook them. Scrape the pot with the silicone spatula to get all the browned bits out.
Dump the flour mixture into the sugar/egg/butter mixture and stir with the spatula until you're no longer seeing big streaks of flour. Add in the chocolate chips, stir to distribute evenly, and spread into the prepared pan.
Bake until the brownies are glossy and just barely firm, about 30 minutes or to an internal temperature of 205°F (96°C). The timing of this recipe is based on a dark metal pan and will vary significantly with glass, ceramic, or light-colored metal.
Cool the brownies to room temperature. Remove by lifting out the parchment paper and cut into 16 squares with a plastic knife. Store in an airtight container with a sheet of parchment paper between layers (it can be the one you lined the pan with).