Preheat your oven to 350°F and lightly grease an 8-inch square baking dish.
In a large bowl, whisk together the flour, 3/4 cup granulated sugar, 1/4 cup cocoa powder, baking powder, and salt.
Stir in the milk, melted butter, and vanilla extract until the batter is smooth.
Pour the batter into the prepared baking dish, spreading it out evenly.
In a small bowl, mix the light brown sugar and 1/4 cup cocoa powder, then sprinkle this mixture over the batter in the dish.
Carefully pour the boiling water over the top, covering the entire surface. Do not stir.
Bake for about 30 minutes, until the center is set and the top begins to crack.
Allow the cake to cool for about 10 minutes before serving. Serve warm, perhaps with a scoop of vanilla ice cream or a dollop of whipped cream.