Preheat the oven to 350. Line the bottom of an 8" or 9" cake pan with parchment paper and spray with bottom and sides with oil. Set aside.
Using a handheld shredder or your food processor with the shredding attachment, shred the carrots. Set the carrots aside and wipe out the food processor. Add the walnuts and mix until they are broken down into uniform pieces.
In a large mixing bowl combine flour, baking powder and soda, spices, and salt. In a separate bowl whisk together the applesauce, oil, white and brown sugars, and vanilla extract. Stir in the grated carrots and coconut. Add the wet ingredients to the dry, gently mixing to combine. Once you have an evenly-hydrated batter, fold in the walnuts.
Pour the cake batter into the prepared cake and bake for about 1 hour, until a knife inserted in the middle of the cake comes out clean. Remove from the oven and let the cake cool for at least 30 minutes before removing it from the cake pan. Let the cake continue to cool on a wire cooling rack.
Once the cake is completely cool, remove from pan by flipping upside down and cut the cake it through the middle into two or three layers.