Start by making the decorations. For the crystallised thyme, put the sugar on a plate, brush the thyme with the egg white, then roll in the sugar. Shake off some of the excess, then leave the sprigs on a piece of baking parchment to dry out completely.
For the candied lemon slices, tip the sugar into a saucepan with 200ml water. Heat gently to dissolve the sugar, then bring the syrup to the boil. Drop in the lemon slices and cook for 15 mins until softened. Carefully remove the slices from the syrup and place on a piece of baking parchment to dry. (Lift them gently as they will be quite delicate, and you want to keep them in rounds if possible.) Keep the remaining syrup for brushing over the cake.
Heat oven to 180C/160 fan/gas 4. Grease and line the bases of two 20cm springform cake tins. Make the four sponges in two batches. To make the first batch, in a stand mixer or in a large bowl using an electric hand whisk, beat 175g butter with 175g sugar and the zest of 1 lemon until creamy. Gently beat three of the eggs together in a jug. Gradually add the egg to the mixture, scraping down the side and whisking well after each addition. In a separate bowl, mix together 200g self-raising flour, 1 tsp baking powder and a pinch of salt, and fold this into the cake mixture. Finally, stir in 300g courgette and 1 tbsp thyme leaves. Divide the mixture between the tins and bake for 25 mins or until a skewer inserted into the middle comes out clean.
While the sponges are still warm, prick all over with a skewer and brush over a little of the reserved lemon syrup. Once cool enough to handle, remove from the tins and place on a wire rack. Make and cook the second batch of cakes using the remaining ingredients – you can use the same baking parchment, just re-grease the parchment and the sides of the tins.
When the cakes are cool, make the icing. Whisk all the ingredients together with a pinch of salt until smooth and fluffy.
Using a cake turntable, if you have one, sandwich the cakes together with a little of the icing. Using a palette knife, spread a thin layer of icing all around the cake and chill for 30 mins to set. (This layer is known as a ‘crumb coat’ because it traps any loose crumbs on the surface of the cake – so when you put a second layer of icing on, the cake will be nice and smooth.) If you are short of time, put it in the freezer for 5 mins.
Once chilled, completely cover the cake in a second layer of icing to achieve a smooth finish. Stick the crystallised thyme and candied lemon slices over one side of the cake and top as if they are cascading.
Return the cake to the fridge or freezer until the icing has completely set. For extra shimmer, use a thin paintbrush to brush a little of the gold paint across the lemon slices for a burnished, autumnal effect, and dot pieces of gold leaf as you go, if you like.