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Place the grated carrot and around three quarters of the orange zest (reserving just enough for the top of the cupcakes, a tablespoon we'd say) into a small bowl.
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In a larger bowl beat together the eggs, caster sugar, oil and most of the orange juice (reserve 3 tbsp's for the icing) until well combined, then mix in the grated carrot and zest.
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Sift the flour, baking powder, nutmeg and cinnamon into the egg mix and gently fold in until all ingredients are combined and thick.
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Place cupcake cases into a cupcake tray, fill each one to about half full, you should get around 18 cupcakes out of the mix.
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Heat an oven to gas mark 5/190c and place the tin in the oven for around 15-20 mins until lovely and golden on the top.
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Remove from the oven and place on a cooling rack. Once cold, mix the icing sugar with a spoonful of orange juice reserved from earlier, you may not need all of it, just add it a bit at a time so you get an icing that is neither to thick or too think.
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Using a teaspoon, place a dollop of icing on the top of each cake, the icing will run nicely down the curves of the cake. Finis

