Preheat the oven to 180℃, brush a quarter sheet pan with oil and set aside.
In a blender or food processor add the carrots and blend until coarse/really fine. Add the eggs, olive oil, sugars and blend again until smooth - take your time in this step we want a smooth texture.
Add the flour, corn flour, baking powder, baking soda, salt, cinnamon and pulse them into the wet ingredients - do not over mix at this stage.
Finish with lemon juice and stir with a rubber spatula. Pour into the prepared tin and bake for 15-17 minutes until done, a tooth pick inserted into the center should come out clean. Let it cool completely before frosting.
To make the frosting beat the butter until pale and light in color around 2-3 minutes. Add the cream cheese and stir together until combined. Add the icing sugar and start beating it slowly, gradually increasing the speed to avoid the sugar going everywhere.
Once combined and smooth, remove 3 spoon fulls in a small bow and 2 spoon fulls in another small bowl. Color the larger portion with orange food coloring and the smaller portion with green food coloring. Hence you will have majority of white frosting, orange frosting and green frosting.
Transfer the oranger and green frosting into piping bags and cut a little of the tip. Frost the cake with the white frosting first, smooth out the layer. Pipe on the carrots with the orange frosting and the carrot tops/leaves with the green frosting. Slice and serve.