Preheat oven to 350°; will need 8-inch square metal baking pan.
In a medium bowl, whisk together flour, 3/4 cup sugar, 2 tablespoons cocoa powder, baking powder, and salt.
Add egg, milk, oil, and vanilla to flour mixture and stir with a wooden spoon until just blended.
Spread batter evenly in prepared pan.
In the same bowl (no need to clean it), stir together brown sugar, the remaining sugar and the remaining cocoa powder.
Sprinkle over batter in the pan.
Slowly and gently drizzle boiling water over batter being careful not to disturb layers (do not stir).
Bake in oven for 40-45 minutes or until top cake layer springs back when touched.
Let cool in pan on a wire rack for 20 minutes.
Scoop into bowls and serve warm with ice cream, if desired.