Preheat the oven to 350°F. For the batter, put the butter, both types of sugar, the eggs, buttermilk, and oil in a large bowl and beat them with an electric whisk until foamy. Then add the other dry ingredients and grate in the carrots and the apples. Mix everything again with a spatula until it's thoroughly mixed together.
Now divide the batter evenly between three large springform pans (diameter: 9 inches) that have been greased with butter and dusted with flour. Bake all three layers for about 25 minutes and then let them cool.
Peel slices from a carrot, until you have peeled about half of it away. Then bring the water and sugar to the boil, add the carrot slices to the syrup and blanch them in it for 1 minute. Scoop them out and lay them next to each other on a baking tray lined with baking parchment. Dry the strips in an oven set to 210°F with the heat coming from above and below for 20 minutes.
Then, one after another, wrap the still-warm strips around the handle of a kitchen spoon, let them cool a little, remove them from the spoon, and collect the curled strips in a bowl.
To prepare the cream: beat the butter and cream cheese together until they are foamy — they should be at roughly the same temperature — then stir in the powdered sugar a little at a time.
Now to assemble your masterpiece. Put the first cake layer on a cake plate and spread several tablespoons of the cream on top of it. Set the second layer of cake on top, cover that with a similar amount of cream and then top it off with the final cake layer. Use the rest of the cream to cover that and round the side of the entire cake.
For decoration, sprinkle with pecan nuts according to taste and then for an eye-catching centerpiece, arrange some curled carrot strips in the middle of the cake.