Preheat convection oven to 325 degrees F or standard oven to 350 degrees F. Grease and lightly flour two 9x1-1/2-inch round baking pans; set aside.
In a large mixing bowl combine flour, granulated sugar, baking powder, baking soda, apple pie spice, and salt. Add carrot, shredded apple, nuts, oil and eggs. Beat with an electric mixer at low speed until combined. Beat 1 minute at medium speed. Pour batter into prepared pans, spreading evenly.
Bake cake layers in convection oven 25 minutes or bake in standard oven 30 minutes or until a toothpick inserted near the center comes out clean. Let cool in pans on a wire rack for 10 minutes. Remove from pans and cool completely.
For frosting, in a food processor bowl place the whipping cream, sour cream, brown sugar, and vanilla. Cover and process for 2 to 2-1/2 minutes or until stiff peaks form, stopping to scrape down sides every 30 seconds. (Or, place ingredients in a large mixing bowl and beat with a hand blender or electric mixer until stiff peaks form.)
To assemble cake, place 1 cake layer on a serving platter. Spread about 3/4 cup frosting on top of the first cake layer. Sprinkle the 1/4 cup crushed apple chips atop frosting. Top with second cake layer; frost top and sides of cake with remaining frosting. Decorate cake with whole apple chips and cinnamon sticks, if desired. Makes 12 servings.
A combination of cream, sour cream, and sugar whips into a wonderfully fluffy frosting for this apple-and-carrot cake.