Heat the oven to 300ºF. Coat an 8-inch square baking dish with non-stick spray and trim the width of a piece of parchment paper so that it fits just inside the baking dish with overhang on two sides. Place it inside the baking dish like a sling.
In a stand mixer fitted with the whisk attachment, beat the eggs at medium speed until fluffy and light yellow, 2 to 3 minutes.
In a small bowl, combine the sugar, brown sugar, cocoa, flour, and salt. With the mixer on low speed, add this mixture to the beaten eggs. When the dry ingredients are fully incorporated, slowly add the melted butter and vanilla extract. Bring the mixture together on low speed.
Pour the batter into the prepared pan and bake for 45 to 50 minutes. Check for doneness with the tried-and-true toothpick method: A toothpick inserted into the center of the pan should come out clean. A probe thermometer inserted into the center should register 195ºF. Let the brownies cool in the pan for 3 minutes, then remove them from the pan using the parchment paper as a sling.
Cut the brownies into 9 pieces using a pizza wheel and then move to a wire rack to cool completely.