Preheat oven to 350°F (175°C). Place the butter in an 8-inch square baking dish and set it in the oven while it preheats, just until the butter is melted.
In a medium bowl, sift together the flour, ¾ cup of the sugar, and 1½ tablespoons cocoa powder. Stir in the milk and vanilla until smooth.
In a small bowl, mix together the remaining 1 cup sugar and ¼ cup cocoa powder.
Carefully remove the baking dish from the oven. Pour the batter evenlv over the melted butter without stirring.
Sprinkle the sugar-cocoa mixture evenly over the batter.
Slowly pour the boiling water over the entire dish. Do not stir.
Bake uncovered for 30 minutes, or until the top looks set and crackled.
Serve warm, spooning the rich chocolate sauce from the bottom over each serving. Excellent topped with vanilla ice cream or whipped cream.