Preheat the oven to 175℃ (350℉).
Grease a rectangle dish or pan with butter. Mine is 27cm x 21cm (10½ inches x 8¼ inches), but you can use any kind of pan (although baking times may need to be adjusted).
Heat the butter until just melted. If warm, set to one side for a minute or two.
In a large bowl or stand mixer, whisk the eggs and sugar together until light and fluffy, and the colour has paled.
Add the butter and vanilla and whisk in.
In a small bowl, combine the coffee powder and milk. Give it a little stir just until the coffee has dissolved.
Add the other ingredients and whisk in.
In a separate bowl, combine the flour, cocoa and baking powder.
Add to the wet ingredients and continue to whisk on a low setting, just until pretty much all of the dry ingredients have been incorporated.
Pour the batter into the prepared pan.
Bake in the middle of the oven for around 20-25 minutes. The cake is ready when an inserted skewer or toothpick comes out clean.
Remove from the oven and allow to cool a little while you prepare the icing.