Preheat your oven to 350 degrees Fahrenheit and grease an 8-inch spring form pan with butter, and line the bottom with a circle of parchment paper.
In a large bowl add the cocoa powder and butter, and pour the hot coffee over them, whisking as the butter melts and the cocoa dissolves in the liquid.
Whisk in the brown sugar, milk and vanilla until the mixture is smooth and well combined.
In a smaller bowl, whisk together the flour, baking soda, baking powder, and salt.
Add the dry ingredients to the wet ingredients and combine them until the mixture is smooth and no streaks of flour remain.
Pour the batter into the prepared pan and bake at 350 degrees Fahrenheit for about 20-25 minutes, or until a toothpick inserted into the centre of the cake comes out clean.
For the frosting:
While the cake cools, add the butter and powdered sugar to a mixing bowl and mix them together until smooth with a hand mixer.
Melt the chocolate in the microwave in a microwave-safe bowl, on 50% powder for 30-second increments just until the chocolate is melted.
Let the chocolate cool for a few minutes before drizzling it into the butter and powdered sugar mixture while the mixer is running.
Continue mixing on medium-high speed as the frosting becomes fluffy, adding the milk as you mix.
Assembly:
When the cake has cooled completely (remove it from the pan after about 10 minutes to a wire rack to cool completely), place it on a plate and spread the frosting over the top of the cake.
Slice and serve.
Video
Notes
This cake keeps well in an airtight container for up to 3 days after baking. It's possible to make this cake in a square 8-inch by 8-inch pan instead of a round pan (choose a pan with high sides if possible, or use a 9-inch square pan).