Preheat the oven to 350 degrees F (175 degrees C). Melt butter in an 8x8 inch baking dish while the oven preheats.
In a medium bowl, stir together the flour, 3/4 cup sugar, and 1 1/2 tablespoons cocoa. Stir in milk and vanilla until smooth. Spoon this batter over the melted butter in the baking dish.
Stir together the remaining cup of sugar and 1/4 cup cocoa powder.
Sprinkle over the batter. Slowly pour boiling water over the top of the mixture.
Bake for 30 minutes in the preheated oven, until set.