Peel and shred the carrots in a food processor. (You can use a box grater, but it takes awhile!) Spray a 9x5" loaf pan with cooking spray and line the bottom with parchment paper. Preheat your oven to 325°.
In a large bowl, whisk together the oil, eggs, sugars, applesauce, vanilla, and carrots.
Whisk or sift together the flour, baking powder, soda, salt, cinnamon, ginger, and nutmeg. Add to the wet ingredients and stir gently, just until combined.
Pour the batter into the prepared loaf pan. Bake at 325° for 55-60 minutes, or until a toothpick inserted in the center of the loaf comes out without raw batter. (A few moist crumbs are fine.)
Let the bread cool in the pan for 10-15 minutes, then carefully dump out and place on a wire rack. Cool completely before frosting. (Or eat the bread without frosting- it's still delicious!)
For frosting: Combine all the ingredients in a small bowl and beat with an electic hand mixer until smooth. I don't ever need to add milk, but you can if you think it's too thick.