Preheat the oven to 325 degrees. Lightly butter an 2-quart (9 by 12 by 2 inch) oblong baking dish.
Melt the butter in a small saucepan and set aside to cool.
In the bowl of an electric mixer fitted with the paddle attachment, beat the eggs and sugar on medium-high speed for 5 to 10 minutes until thick and yellow.
Meanwhile, sift the flour and cocoa powder and set aside.
When the egg mixture is ready lower the speed and add the vanilla, cocoa, and flour mixture. Mix only until combined. With the mixer on low, slowly pour in the cooled butter and mix again just until combined.
Pour the brownie mixture in the prepared dish and place into a larger baking pan. Add enough of the hottest tap water to the pan to the halfway up the side of the dish and bake for exactly 1 hour. A cake tester inserted 2 inches from the side will come out three quarters clean. The center will appear very under-baked; this dessert is between a brownie and a pudding.
Allow to cool and serve with vanilla ice-cream.