In a medium saucepan over medium heat, melt the butter. Continue cooking, swirling the pan occasionally, until the butter foams and turns golden brown with a nutty aroma and dark specks on the bottom, about 4 to 5 minutes. Immediately pour into a large heatproof bowl, scraping in all the browned bits. Let cool 5 minutes.

Add the chopped chocolate to the warm brown butter and let sit 2 minutes, then whisk until completely smooth. If needed, microwave in 15-second bursts until melted.

Whisk in both sugars and the espresso powder until well combined. The mixture will look thick and glossy.

Add the eggs one at a time, whisking vigorously after each until fully incorporated. Whisk in the vanilla. The batter should be smooth and pull away from the sides of the bowl.

Switch to a rubber spatula. Add the flour, cocoa powder, and salt. Fold just until no dry streaks remain; do not overmix.

Pour batter into a parchment-lined 8x8-inch pan, spreading to the corners. Bake at 325°F until the top is set and a toothpick inserted in the center comes out with moist crumbs, 30 to 35 minutes. Cool completely in the pan on a wire rack before lifting out and cutting into 16 squares.