Preheat the oven to 350° F. Line a 9x9-inch baking pan with aluminum foil, and spray the foil with cooking spray.
Melt the butter with the unsweetened chocolate in the microwave, stirring occasionally, until the butter and chocolate are entirely melted. Whisk to combine.
Add the sugar and cocoa powder and whisk until they're incorporated. At this point the mixture will look grainy and not very appetizing. Add the eggs one at a time, being sure to whisk well after each addition. With each egg added, the mixture will get smoother and shinier, and after all 3 have been added and the batter has been well-mixed, the graininess should be gone and your brownie mixture should have the texture of chocolate pudding.
Whisk in the vanilla extract, then stir in the flour and fine sea salt until any dry streaks disappear.
Pour the batter into the prepared pan and smooth the surface with a rubber spatula. Sprinkle the remaining large-flaked sea salt over the top of the brownies.
Bake for about 35 minutes, until the edge is set but the center is still a bit soft and a toothpick inserted into the center comes out coated with a little of the batter. Cool at room temperature in the pan for about an hour, then refrigerate just until they are firm.
Lift the brownies from the pan and remove foil. Cut the brownies 16 squares, and just before serving, sprinkle with just a bit more salt for visual appeal.