Mix flour, white sugar, 2 tablespoons cocoa powder, and baking powder together in a bowl. Add milk, oil, and vanilla extract; mix thoroughly. Pour batter into an ungreased 9-inch baking pan. Sprinkle brown sugar and 1/4 cup cocoa powder on top. Pour hot water on top immediately before baking.
Bake in the preheated oven until a cake layer forms on top and springs back when lightly pressed, 30 to 40 minutes.