Preheat the oven to 350°F.

Spray two 9-inch cake pans with canola oil cooking spray and set aside.

Using an electric mixer, gently cream the butter with the oil.

Add the applesauce and the sugar, beating until well combined.

In a medium bowl combine both flours with the baking soda, cinnamon, and salt.

Add the dry ingredients, alternating with the eggs, beating well after each addition.

Add the vanilla extract. Add the carrots and beat on medium speed until well incorporated, about 2 minutes.

Fold in the nuts.

Divide batter between the cake pans and bake for about 30 to 40 minutes or until done when tested with a toothpick or thin knife.

Remove from the oven and let the cakes rest in the pans for 10 minutes. Invert onto wire racks, remove from the pans, and let cool.

For the frosting, blend softened cream cheese with sugar, milk and vanilla.

Frost the cakes with desired amount of frosting and serve. Refrigerate, covered, for up to 3 days.