Position a rack in the middle of the oven and preheat to 350°F.
Grease a 9x13-inch dark metal pan with softened butter, then line with parchment paper, leaving overhang on all sides. Grease the parchment with softened butter. You could also use cooking spray if you don't want to mess with smearing butter, but butter will give the best result.
Combine the chocolate, ¼ cup of cocoa powder, and espresso or espresso powder in a heatproof liquid measuring cup or medium bowl and set aside.
Add the butter to a medium saucepan over medium heat and cook until the butter just comes to a vigorous simmer, continue cooking over medium heat for approximately 6-8 minutes, stirring continuously with a wooden spoon until the butter turns a golden brown and smells nutty and rich. The butter will foam and sizzle, you'll see the color come through the foam. Watch closely so the butter doesn't burn!
Immediately pour the hot browned butter over the chocolate mixture and whisk until the chocolate is completely smooth and melted, then set aside for a few minutes to cool.
Combine the granulated sugar, brown sugar, vanilla extract, and eggs in a large bowl. Beat with an electric hand or stand mixer on high speed until light and fluffy, about 10 minutes. It will be similar to the texture of extra fluffy pancake batter.
With the mixer on, pour in the slightly cooled chocolate and butter mixture and blend until smooth.
Sift in the flour and remaining ½ cup cocoa powder and use a rubber spatula to gently fold until just combined. This will take some patience in folding.
Pour the batter into the prepared baking pan and smooth the top with a spatula.
Bake at 350° F until lightly puffed on top, about 20 minutes.
Remove the baking pan from the oven using oven mitts or kitchen towels, then lightly drop the pan on a flat surface 2-3 times until the brownies deflate slightly. Sprinkle with the sea salt if desired.
Return the pan to the oven and bake until a toothpick inserted into the center of the brownies comes out fudgy but the edges look cooked through, about 15 minutes more. The center of the brownies will seem a little under-baked, but that's because they are so fudgy!
Set the brownies on a cooling rack and cool completely in the pan.
Use the parchment paper to lift the cooled brownies out of the pan. Cut into 12 big brownies or 24 small brownies and enjoy!