Photograph: Karen Thomas
Preheat the oven to 180°C, fan 160°C, gas 4. Sift the flour, baking powder and spices into a bowl. Add the eggs, sugar and oil; whisk.
In another bowl, mix the carrots, pistachios, sultanas and lime zest. Fold into the egg mixture. Divide the mixture between three greased and lined 20cm x 4cm sandwich tins. Level with the back of a spoon, then bake on the middle shelf for 25 minutes.
For the frosting, whisk the butter for a minute. Add the other ingredients; combine.
Cool the cakes, then turn out on to a wire rack. Layer the cakes with the frosting, then coat the outside. Dust with cinnamon and top with blueberries and lime zest.
Kitchen secret: it's important to use real Philadelphia cream cheese for the icing, as it contains natural stabilisers that help give a thick and fluffy icing. Don't be tempted to use low-fat versions or 'soft white cheese' instead