Preheat your oven to 350°F (175°C) and lightly grease an 8x8-inch baking dish.
In a medium bowl, whisk together the flour, 1/2 cup of granulated sugar, 1/4 cup cocoa powder, baking powder, and salt.
Stir in the milk, melted butter, and vanilla extract until just combined and smooth. Do not overmix.
Spread the batter evenly into the prepared baking dish.
In a separate small bowl, mix together the remaining 1/4 cup granulated sugar, brown sugar, and the second 1/4 cup of cocoa powder. Sprinkle this dry mixture evenly over the top of the batter.
Pour the boiling water gently over the top of the entire mixture. Do not stir.
Bake for 40 to 45 minutes, or until the cake portion is just set and slightly jiggly in the center.
Remove from the oven and let cool for 15 minutes before serving warm to allow the pudding layer to thicken slightly.