Unlike cakes that go into the pan as a single batter, this one has two additional layers: a sugary cocoa topping, followed by a generous puddle of boiling water. As the cake bakes, the water heads south to form a chocolaty pudding at the bottom.
Generously grease a 9-inch square cake pan with butter. (You can also use a 9 1/2-inch deep-dish pie pan, provided that the sides measure at least 2 inches high.) Heat the oven to 350º F.
In a large bowl, sift together the flour, sugar, 2 tablespoons of the cocoa powder, the baking powder, and the salt.
In a medium-size bowl, stir the remaining 2 tablespoons of cocoa powder into the brown sugar. Rub the mixture with your fingers, breaking up any sugar lumps, until evenly combined. Set the bowl aside.
Make a well in the flour mixture. Add the egg, milk, melted butter, and vanilla extract. Whisk the liquids to blend them, then briskly stir the entire mixture, pulling the dry ingredients in from the sides to make a uniform batter.
Scrape the batter into the buttered pan and spread it evenly with a spoon. Sprinkle the chocolate chips over the batter. Then spread all of the brown sugar mixture evenly over the top.
Slowly pour the boiling water over the batter. Place the pan on the center rack and bake the cake for 30 minutes. When it's done, the cake will be baked through, but there will be a thick layer of chocolate pudding at the bottom. Because this is difficult to check with a toothpick, it's best to rely on time to determine when the cake is finished.
Transfer the cake to a wire rack and let it cool for about 10 minutes before serving. You can cut the cake into squares or slices, or simply spoon it out of the baking dish. Serve topped with whipped cream or ice cream. Makes 6 to 8 servings.