Preheat your oven to 180 degrees C and line a 7x7inch/20x20cm baking tin or loaf tin with parchment paper.
Grate the courgette on the coarse setting of a box grater (too small and the courgette will just turn to mush!) and set aside.
Make your flax egg in a small bowl by combining 2 tablespoon flax seed with 5 tablespoon of water, and set aside for the flax seeds to soak up the water.
In a large mixing bowl, add all the dry ingredients and stir to combine.
In a separate bowl, combine the vanilla essence, apple sauce, coconut oil, brown sugar, caster sugar, lime zest, and lime juice.
Add the wet ingredients and the courgette to the dry ingredients and stir until just combined - the batter will be fairly thick!
Pour into your lined baking tin and smooth out the top of the batter.
Bake the cake for 35 - 45 minutes, or until a skewer comes out clean when inserted.
Leave to cool for 10 minutes before removing the cake from the tin and allowing it to cool fully on a baking rack before icing. If you can, refrigerate your cake for an hour before icing.
To make the icing:
Beat the cream cheese and butter together on a low setting
Add the lime zest and about one-third of the icing sugar, and beat
Finally, mix in the remaining icing sugar for a fluffy cream cheese frosting!
Once you've made the icing, slather it over the top of your cake and scatter over the remaining lime zest.