Preheat the oven to 350 degrees. Place in the oven a roasting pan large enough to hold the ovenproof dish. Fill it one-quarter full of water.
Place the butter and 1 cup of the brown sugar in the bowl of an electric mixer. Using the paddle attachment cream on medium high speed for 2 minutes, until light. Continue to mix, slowly add the eggs and then the vanilla extract.
Sift together the flour, salt, baking soda, and the ⅓ cup of cocoa powder. Increase to low speed and stir the dry ingredients into the creamed mixture. Stir in the 2 tablespoons of water. Spread the batter in the ovenproof dish.
In a small bowl, mix together the remaining ½ cup brown sugar, 1½ teaspoons cocoa powder, and the 6 tablespoons hot water. Pour the cocoa liquid over the top of the chocolate batter.
Put the pudding in the water bath in the oven. The water should come halfway up the sides of the baking dish. Add or take out water as necessary.
Bake the pudding for 55 to 60 minutes. When done, the top of the pudding will be cracked and will have pulled away from the sides of the dish. A skewer inserted in the middle of the pudding should come out clean.
Remove the pudding from the oven and sprinkle the top with the 3 tablespoons of powdered sugar. Serve warm with the heavy cream on the side.