Make the Brownies: Preheat your oven to 350° F. Line a metal 9×9 baking dish with parchment paper. Add the sugar and eggs to a large mixing bowl. Beat with an electric mixer on high speed for 3-4 minutes until the sugar is dissolved (rub the mixture between your fingers – if you feel granules of sugar, beat it a bit longer, until it feels almost completely smooth). Whisk in the melted butter and vanilla. Add the flour, cocoa powder, and salt, and fold in until you have a thick batter.
Bake the Brownies: Pour the batter into your baking dish and bake for 33-38 minutes. The top will look set, and a toothpick inserted into the center will have thick, moist crumbs on it, but it shouldn't have thick, gooey batter. A toothpick inserted into the edge will come out mostly clean (You can also use the tip of a knife to lift up a little piece from the center to more clearly see the doneness, if you have a hard time judging when brownies are done). Don't over bake these! They will set up as they cool. I promise.
Serve + Store: Let brownies cool for at least 1 hour before slicing. They're amazing warm, but won't hold a neat shape and will appear under done if you cut into them immediately. The longer they cool, the more they will set up to the proper fudgy consistency. Store cooled brownies in an airtight container at room temp for 3-4 days.