Set the oven to 350°F (175°C) and grease a 9x9-inch baking dish.
In a bowl, whisk together flour, 1/2 cup cocoa powder, sugar, baking powder, and salt.
Stir in milk, melted butter, and vanilla until smooth. Spread the batter evenly in the dish.
In another bowl, combine brown sugar and the remaining 1/4 cup cocoa. Sprinkle over the batter without mixing.
Gently pour boiling water over the top. Do not stir—this creates the fudge sauce underneath.
Place in the oven and bake for 30–35 minutes, until the top is set and the center feels firm.
Scoop into bowls, making sure to get the cake and sauce. Serve warm with ice cream or whipped cream.