Preheat the oven to 160c/gas3. Line the base of a springform 8in cake tin with greaseproof paper. With a electronic whisk cream together the butter both types of sugar until light & fluffy. Beat in the eggs one at a time until well combined.
Sift together all the dry ingredients & using a wooden spoon stir them into the creamed mixture.
In a seperate bowl, combine the carrot, apple & chopped nuts.Stir in the apple juice & vanilla extract, then fold the carrot mixture into the cake mixture.Spoon into the tin & bake in the center of the oven for 1 hour 35 minutes, or until deep golden brown & a stewer inserted into the cake comes out clean.
Meanwhile, to make the frosting , combine the cream cheese, unsalted butter & orange juice together with a wooden spoon. Gradually incorporate the icing sugar into the mixture, beating well until well combined.
Cool the cake on a wire rack, then remove from the tin. Place the cake on a serving plate. Using a palette knife, cover evenly with frosting & sprinkle over the chopped nuts & zest.