Preheat oven to 350 degrees.
SIFT together in a large bowl flour, 1/3 cup cocoa, salt, and baking powder. In a medium bowl whisk together sugar, milk, eggs, butter, and vanilla and stir into flour mixture until combined well. Spread batter into an 9-inch metal pie plate.
WHISK together in another medium bowl remaining 1/3 cup cocoa, brown sugar, and 1 1/2 cups boiling water. Pour evenly over batter. (This may look strange at this point, but don’t worry. During baking cake forms on top and pudding underneath.)
BAKE until a toothpick inserted in the center comes out with crumbs adhering to it, 30 to 35 minutes. Cool slightly and serve warm with ice cream or whipped cream.