Preheat the oven to 190°C/gas mark 5. Line a 20cm x 30cm baking tray with baking parchment.
Whisk together the eggs, oil and milk. Add the brown sugar and mix well. Sift the flour, cocoa and baking powder into a bowl, then fold into the wet mixture. Finely grate the courgettes and stir into the mixture along with 1 tsp vanilla extract. Spoon into the baking tray and bake for 40 minutes, then leave to cool.
Meanwhile, beat the butter, vanilla extract and 250g icing sugar in a bowl. Spread the icing over the top of the cooled cake and finish with a dusting of cocoa. Cut into slices, remove from the tray and serve.