Preheat oven to 350 degrees F. Line the bottom of a 8x4" bread pan with a piece of parchment paper then spray lightly with cooking spray.

In a large mixing bowl, combine sugars, eggs, applesauce, oil, and vanilla, mixing well to combine. Stir in shredded carrot.

In a separate mixing bowl, combine flour, baking soda, baking powder, salt, cinnamon, and allspice. Add to wet mixture then stir to combine.

Bake at 350 degrees for 45-55 minutes, until a toothpick inserted in the center comes out clean. (If the loaves are browning too much on top during baking, tent a piece of foil over top until they're finished baking).

Cool in pan for a few minutes before removing to a wire cooling rack to cool completely.

Make Frosting (optional): Beat cream cheese and butter together until smooth, about 2 minutes. Mix in vanilla then add powdered sugar, one cup at a time, mixing well. For best results, beat frosting for several minutes, to make it extra smooth and fluffy/creamy in texture. Spread on top of cooled loaf, slice, and enjoy!