Beat sugar, eggs, vanilla and cooled melted butter (cooled so it won't cook the eggs) in a large mixing bowl by hand. You don't need an electric mixer for this batter.
Add self-rising flour and cocoa powder to the bowl. Mix well. This is a very thick batter.
Pour into a greased or use parchment paper lined 8 inch square baking pan. Spread evenly.
Place in 350°F oven for 25 - 30 minutes or until tested with a toothpick that comes out with a few crumbs on it. You don't want to over bake the brownies.