Melt butter, milk, and coffee together in a saucepan, then let this mixture cool for about 10 minutes. Preheat your oven to 350 degrees F (175 degrees C)
Add the eggs and sugar to a bowl or stand mixer (with the whisk attachment) and whisk together until light in color, fluffy, and about doubled or tripled in size. When you lift the whisk out of the mixture, you should see the mixture flow on top of the surface and sit there for just a moment before dissolving back in - this is called "ribbon stage!"
Add the vanilla extract to the egg/sugar mixture and whisk to combine.
Sift the dry ingredients - flour, cocoa powder, baking powder, and salt - together in a separate bowl. Cocoa powder tends to get lumpy so it is important to sift for a smooth batter!
Add half of the wet ingredients (butter, milk, and coffee) to the fluffy egg and sugar mixture. Whisk to combine, for just about 10 seconds.
Then, add half of the dry ingredients and whisk to combine, for probably around 15-20 seconds.
Add the rest of the wet ingredients, mix, and then add the rest of the dry ingredients and mix until there are no more dry spots.
Give the whole mixture a few folds with a rubber spatula to make sure you've incorporated all the dry ingredients.
Prepare a baking tin by buttering the bottom and sides and lining with parchment paper. You can use either a round cake tin (9-10 inch/22 cm), a square cake tin (9x9 inch/22x22 cm), or a rectangular cake tin (11x7 inch/28x18 cm) - whichever you have on hand or prefer.
Pour the batter into the cake tin, pressing it all the way to the edges, and smooth out the top.
Bake for 25-30 minutes, or until a cake tester emerges clean from the center of the cake. Let this cool fully on a wire rack, removing it from the tin, before decorating.
Once the cake has cooled, make the icing by combining the powdered sugar, cocoa powder, and coffee until you have a smooth, thick icing. You can add more powdered sugar or water to adjust the consistency to your preference.
Pour the icing over the cake and spread into an even layer.
Top with desiccated/shredded coconut. Serve and enjoy!