Heat the butter in a saucepan over medium heat, cutting it into small cubes to speed the process. Stir with a rubber spatula as it melts, foams, and transitions from opaque to translucent. Pull it off the heat when the milk solids at the bottom are golden brown and the butter smells nutty.

Pour the brown butter immediately over the chopped chocolate and kosher salt in a heat-safe bowl, scraping every brown bit from the bottom of the pan. Let the butter sit over the chocolate for a few minutes, then whisk until smooth and fully combined. Set aside to cool to roughly body temperature.

Preheat the oven to 350°F conventional or 325°F convection. Line a 9x9 baking pan with two pieces of parchment criss-crossed so they come up all four sides.

Combine the eggs, vanilla, sugar, and brown sugar in a stand mixer fitted with the whisk attachment. Mix on medium to combine, then increase to high and whip until the mixture reaches ribbon stage, falling in thick, pale ribbons that hold their shape briefly on the surface.

Confirm the chocolate butter mixture is at body temperature, then slowly pour it down the side of the mixing bowl with the mixer running on medium. Keep the stream away from the whisk to prevent splashing.

Remove the bowl from the mixer and fold several times with a rubber spatula to ensure the chocolate and egg mixture are fully combined, reaching the bottom of the bowl where the whisk may have missed.

Sift in the cocoa powder and flour, then fold until just incorporated with no dry streaks remaining.

Pour the batter into the prepared pan and spread evenly with an offset spatula. Bake for about 20 minutes, until the crackle top has spread fully to the center and the middle is puffed rather than sunken. Cool completely before cutting.