Preheat the oven to 180°C, gas mark 4, and grease and line 3 x 20cm sandwich cake tins.
Using a stand mixer ﬁtted with a paddle attachment, or a hand mixer, cream the butter and sugar until very light and ﬂuffy – about 8 minutes.
Add the eggs one at a time, beating well in between each addition. Add the vanilla.
Sift the ﬂour and bicarbonate of soda together and fold into the cake batter.
Divide the batter between 3 medium-sized bowls and stir the grated and puréed vegetables into each bowl (pea and courgette in one, butternut in the second and beetroot in the last bowl).
Spread each bowl of batter into a prepared cake tin and bake in the preheated oven for about 40 minutes or until a skewer inserted into the middle of the cakes comes out clean.
Allow the cakes to cool in the tins slightly, then remove and cool completely on a cooling rack.
To make the icing, preheat the oven to 200°C, gas mark 6, and bake the vanilla pod for 5 minutes or until completely dried. Allow to cool.
Blitz the dried vanilla pod in a food processor or blender until a ﬁne powder forms.
Cream the icing sugar and butter until light and ﬂuffy – about 8 minutes – then whip in the cream cheese, lime zest, lime juice and vanilla until just combined.
Place in a piping bag ﬁtted with a medium-sized straight nozzle.
To assemble the cake, place a cake layer on a serving plate or cake stand, pipe dollops of frosting on top.
Repeat the layers with the cake and frosting, ﬁnishing with a layer of frosting on top.
Decorate the ﬁnal layer with edible ﬂowers.