Preheat oven to 180 C / Gas 4. Grease an 20cm cake tin.
Combine butter and chocolate in a microwave-safe bowl; heat in the microwave until melted, 20 to 30 seconds. Stir milk and vanilla extract into chocolate mixture.
Mix flour, 150g caster sugar, baking powder and salt together in a separate bowl; stir in chocolate mixture until cake batter is smooth. Pour batter into the prepared cake tin.
Stir coffee, brown sugar, 100g caster sugar and cocoa powder together in a bowl; pour over cake batter.
Bake in the preheated oven for 30 minutes. Cool cake in the tin for 10 minutes. Turn cake out onto a serving plate. (Be sure to have a plate or platter that fits tightly over the cake tin before you flip it. The topping will be runny until it cools more).