Preheat the oven to 350°F. Grease an 8x8-inch baking dish. In a mixing bowl, combine flour, baking powder, salt, sugar, and cocoa powder. Stir in milk, melted butter, and vanilla extract until a smooth batter forms. Spread the batter evenly into the prepared dish.
In a separate bowl, mix brown sugar and cocoa powder. Evenly sprinkle this mixture over the batter, ensuring it covers the entire surface. This will create the gooey chocolate sauce as the cobbler bakes.
Carefully pour hot water over the batter and topping. Do not stir. The water will sink into the mixture, forming the rich chocolate sauce beneath the cake as it bakes.
Place the dish in the oven and bake for 35–40 minutes, or until the top is set but the bottom remains soft and saucy. The cobbler will firm up slightly as it cools.
Let the cobbler rest for 10 minutes before serving. Scoop into bowls and serve warm with vanilla ice cream for the perfect combination of hot and cold textures.