Heat your oven to 325 degrees F, butter an oval or square casserole dish.
Sift your cocoa powder, espresso powder, and flour together and set aside. Mellt your butter and set aside.
Using a hand mixer, beat your eggs and sugar together on medium high speed for 5-7 minutes minutes until very thick and light yellow.
Add your vanilla and your dry mixture.
Pour in your melted butter and mix until combined, pour in your buttered casserole dish.
Place your casserole dish in a deep, oven safe baking pan and carefully pour warm water into the pan until it comes up halfway up the side of the casserole dish. This is your water bath to help keep the brownie pudding liquid.
Bake for 60 minutes.
Take out and let cool. It will look underdone but it is in fact ready and good to eat. You can serve warm, or completely cooled. Serve with vanilla ice cream, whip cream and fresh fruit if desired. If served warm or without setting in the fridge, brownie pudding will be softer and runnier. If you allow it to cool for several minutes or hours, it will thicken significantly.
