Preheat the oven to 350°F (175°C). Lightly grease an 8x8-inch baking dish.
In a mixing bowl, whisk together the flour, granulated sugar, cocoa powder, baking powder, and salt.
Add the milk, melted butter, and vanilla extract to the dry ingredients. Stir to combine until you form a smooth batter.
Pour the batter into the prepared baking dish and spread it out evenly.
In a small bowl, mix the brown sugar and cocoa powder for the topping. Sprinkle it evenly over the batter. Do not stir.
Pour the hot water evenly over the top of the batter. Do not stir.
Bake for 35-40 minutes until the top is set, but the bottom remains gooey and fudgy.
Let the cobbler cool for 5-10 minutes, allowing the sauce to thicken slightly.
Serve warm with vanilla ice cream or whipped cream.