Line an 8x8 inch baking pan with parchment paper. Leave overhang on both sides, so that it's easy to remove the brownies from the pan when finished. Don't preheat the oven just yet.
Combine chopped chocolate and one tablespoon of cocoa powder in a DRY bowl. Melt unsalted butter on the stove or in the microwave in 30-second intervals. The butter should be hot. Pour it over the chocolate and let sit for about a minute or so. Then, stir with a whisk until the chocolate has completely dissolved in the butter. It will look silky and shiny.
Preheat the oven to 350 degrees F.
Combine the salt, sugars, vanilla extract, and one egg in a large bowl. Begin beating, while adding the remaining eggs one by one. Beat on medium-low speed for at least 10 minutes, until the mixture is thick and lighter in color (resembling pancake batter).
Ensure that your melted chocolate is cool before doing this step! Pour the chocolate mixture into the egg-sugar mix. Beat on medium-low speed until smooth and combined, about 1-2 minutes.
Sift in the flour and the remaining tablespoon of cocoa powder. Using a rubber spatula, fold in the dry ingredients gently until just combined. Don't overmix.
Pour the batter into the prepared baking pan. Bake for about 15-18 minutes, or until the top has risen and is puffed up. Remove from the oven and lightly drop onto a flat surface 1-2 times. The top will deflate slightly. Sprinkle with coarse sea salt, if using.
Return to the oven and bake for about 15-20 minutes more. The brownies are finished when a toothpick inserted in the middle comes back mostly clean with fudgy crumbs. The edges will look finished, but the center may look underdone; that's what we want! Do not over-bake.
Allow the brownies to cool on a cooling rack for no less than an hour. When they are cool enough, cut into squares and serve.