Preheat the oven to 350°F (175°C) and lightly coat a square baking dish, approximately 9x9 inches in size, with butter or nonstick spray.
Begin preparing the batter by combining the all-purpose flour, baking powder, salt, granulated sugar, and cocoa powder in a mixing bowl. Stir in the milk, melted butter, and vanilla extract until a smooth, cohesive mixture forms. Spread this batter evenly into the greased baking dish.
In a separate small bowl, blend together the brown sugar, remaining granulated sugar, and cocoa powder to form the dry topping. Distribute this mixture uniformly across the surface of the batter without mixing it in.
With care, pour the boiling water over the entire surface of the batter and topping. Do not stir; allow the water to settle naturally to create the sauce as it bakes.
Transfer the dish to the oven and bake for 35 to 40 minutes. The top should appear firm and cracked, with visible bubbling chocolate sauce around the perimeter.
Once out of the oven, allow the cobbler to sit for 10 to 15 minutes. Serve warm, ideally paired with a scoop of vanilla ice cream or a dollop of whipped cream for an elevated finish.