Preheat the oven to 325 degrees F. Lightly butter a 9x13 baking dish. Note: You will need a slightly bigger dish to place this dish in while you cook it with a water bath so keep that in mind. I placed a 9x13 inside a 15X17 baking dish. Note: A water bath gives the best result but is not absolutely necessary.
Melt the 1 cup of butter 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 2-3 minutes, until very thick and light yellow. Meanwhile, sift the cocoa powder and flour together and set aside (I'm very guilty of never sifting).
When the egg and sugar mixture is ready, reduce the speed to low and add the vanilla, and the cocoa powder and flour mixture. Mix only until combined. With mixer still on low, slowly pour in the cooled butter and mix again just until combined.
Pour the brownie mixture into the prepared dish and place it in a larger baking pan. Add enough of the hottest tap water to the pan to come halfway up the side of the dish and bake for 50-60 minutes. A cake tester inserted 2 inches from the side will come out 3/4 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.
Notes
adapted from Ina Garten ; via my friend Jenn Neville