Preheat the oven to 180C/350F gas 4, then grease a 20cm tin with the butter and line with baking parchment.
Grate the courgettes coarsely (there is no need to peel them); if they are grated too finely they will turn to mush.
Put the eggs, oil and sugar in a bowl then beat by hand until creamy.
Fold in the flour, baking powder and salt. Stir in the courgette and the zest. The mixture does look unusual at this stage, but bear with it as it completely transforms when baked!
Pour the mixture into the prepared tin and bake for 35–40 minutes, until golden on top and firm to touch or a skewer inserted comes out clean. Remove from the oven and leave to cool completely in the tin before removing and decorating.
To make the icing, beat the cream cheese, icing sugar and 1/2 the lime zest together until smooth. Spread on the top of the cooled cake.
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