Preheat SC to high. Grease a 20cm (base) round cake pan lightly with oil, and line with non-stick baking paper. Peel and grate the carrots, and set aside. Sift the flours, bicarbonate of soda and cinnamon into a large bowl.
Put the brown sugar, oil, golden syrup, eggs and vanilla in a separate bowl. Use a balloon whisk to mix until combined.
Pour the oil mixture into the dry ingredients. Use a wooden spoon to stir gently until just combined. Stir in the grated carrot.
Pour the mixture into the pan and pop in SC with a rack or jar lid underneath to keep pan from touching base of SC. Place a tea towel under lid and cook for 2hr 15 mins (this time will vary depending on your Sc, I would advise checking every 10-15 mins after about 1 1/2hrs. Set aside for 5 minutes, before turning out onto a wire rack to cool completely.
To make the icing, place the cream cheese, icing sugar and vanilla in a bowl. Use a wooden spoon to mix until well combined.
Spread the icing over the cake.