Line a 20cm x 30cm baking tray with baking parchment and preheat the oven to 190°C, fan 170°C, gas 5.
In a bowl, whisk the eggs, oil and milk together, 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 teaspoon vanilla extract. Spoon into the baking tray and bake for 40 minutes. Leave to cool.
To make the icing, beat the butter, 1 teaspoon 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.
Cook's tip: If you don't have a baking tray, you could also cook this cake in 2 x 18cm round cake tins. Simply reduce the cooking time to 30 minutes.