1. Preheat oven to 350°F (180°C).

2. Grease and line two 9-inch (23cm) round cake pans with parchment paper.

3. In a medium bowl, whisk together flour, baking powder, baking soda, cinnamon, cardamom, ginger, and salt. Set aside.

4. In a large bowl, beat butter and both sugars with an electric mixer until light and fluffy, about 3 minutes.

5. Add eggs one at a time, beating well after each addition. Mix in vanilla.

6. Stir in grated carrots and apples until evenly distributed.

7. Gradually add the flour mixture, mixing on low speed just until combined. Do not overmix.

8. Divide the batter evenly between the prepared pans and smooth the tops with a spatula.

9. Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.

10. Let cakes cool in pans for 10 minutes, then turn them out onto wire racks to cool completely.

11. For the frosting, beat softened butter and cream cheese together until smooth and creamy.

12. Gradually add granulated sugar, beating until light and spreadable.

13. Place one cake layer on a serving plate and spread half the frosting on top.

14. Add the second layer and spread the remaining frosting over the top and sides.

15. Serve at room temperature.