Preheat your oven to 350 degrees F. Spray three 8-inch round cake pans with nonstick spray, line the bottoms with parchment paper and spray again. Set aside.
In a medium size mixing bowl, combine the flour, spices, baking powder, baking soda and salt. Whisk to combine. Set aside.
In the bowl of an electric mixer fitted with the paddle attachment, combine the oil and sugars. Beat on medium speed for about two minutes.
Add the applesauce and vanilla and then stir to combine.
With the mixer on medium speed, gradually add the eggs, making sure to scrape down the sides and bottom of the bowl between additions. Once all of the eggs are combined, beat the mixture for about a minute on medium speed.
With the mixer on low speed, gradually add the flour mixture.
Fold in the carrots.
Evenly divide the cake batter between the three prepared pans (about 20 ounces of batter in each). Bake for 25 to 27 minutes, or until a toothpick inserted in the center comes out with a few moist crumbs on it.
Let the cake layers cool in the pans on cooling racks for about 10 to 15 minutes. Invert the cake layers onto the cooling racks to cool completely.
Level cake layers, if needed.
Cake layers can be made ahead of time. To store: let cake layers cool to room temperature and then wrap well with plastic wrap and freeze. Before you decorate your frozen cake layers, let them thaw at room temperature for about 30 to 60 minutes prior to decorating.