Preheat oven to 350 F. Combine flour and salt in a bowl and set aside. In a medium saucepan, melt the chocolate and the butter while stirring frequently. Add 1 cup of the sugar and stir for about 30 seconds then turn off the heat. Stir in the vanilla extract. Transfer this mixture to a large mixing bowl.
Combine the remaining 1 cup of sugar and the eggs in the bowl of your stand mixer (you can also use a handheld mixer). Whisk them using a hand whisk for now, just to combine.
Slowly pour half of the egg-sugar mixture to the chocolate mixture. Continuously whisk the chocolate mixture while pouring the egg mixture in. This is important so that the eggs don't set from the heat of the chocolate mixture.
Now use the whisk attachment of the stand mixer (or use a handheld mixer) to beat the remaining egg-sugar mixture until it is thick, pale and doubled in volume. Use a rubber spatula to fold this mixture into the chocolate batter until mixture is uniform. Then fold in the flour mixture until incorporated.
Pour batter into an ungreased 9-inch square glass or ceramic baking pan. Bake for about 28 minutes. The brownies should be just about set with some little cracks on the edges. Let cool completely before cutting into squares.
Adapted from Baking with Julia by Dore Greenspan