Start by grating the zucchini. Once it is grated, squeeze the zucchini to remove excess moisture. Set this aside, it will be the last thing mixed in. Wash your lemon well with soap to clean it thoroughly then dry.
Zest and juice the lemon(s). Mix just the zest and sugar in a small bowl. Mix the lemon sugar with the melted butter. To this, mix in the egg, vanilla, and lemon juice.
Top tip: Be sure not to overmix the batter once the egg goes in. This will result in a tough cake. It is important to only mix until ingredients are incorporated, nothing more.
Mix the flour, baking soda, and salt in a separate bowl. Add this to your batter and mix to incorporate. Last in is the grated and squeezed zucchini. Mix this into your batter without overmixing.
Pour into your prepared loaf pan and bake until a toothpick comes out clean. Remove it from the oven and let it cool for 5 minutes before turning it out onto a cooling rack.
Once the cake is cooled it is time to top it with the lemon cream cheese icing. Mix the softened cream cheese, lemon zest, lemon juice, powdered sugar, and vanilla into your bowl. Mix with a stand mixer, hand mixer, or even by hand if you prefer. Once it is creamy and smooth, spread it over the top of the zucchini cake.