Set out eight tablespoons of butter and eight ounces of cream cheese to soften. Preheat the oven to 350 degrees F. Grease and flour a 9x13 pan (see notes for other pan sizes.)
Rinse the carrots, peel them with a vegetable peeler, and cut off the ends. Shred the carrots using a food processor or grater and set them aside.
In a large mixing bowl, whisk the eggs, oil, applesauce, water and vanilla until combined. Whisk in the sugar until mixed evenly.
In another bowl, stir together the flour, cinnamon, allspice, salt, baking soda and baking powder. Stir those dry ingredients into the wet ingredients in your large bowl, just until moistened. Gently fold in the carrots (using an under-and-over motion) with a spatula, until they are evenly distributed.
Transfer the batter to your pan and bake for 30 minutes, or until a toothpick inserted in the center comes out with a few moist crumbs. Let the cake cool completely before frosting it.
For the frosting, in a large bowl, beat the softened cream cheese, butter and vanilla with an electric mixer until light and creamy. Gradually beat in the powdered sugar until smooth.
In a separate (ideally stainless steel) bowl, and with clean, dry beaters, beat the heavy whipping cream until stiff peaks form. Fold the whipped cream into the frosting and then whip to combine it, just until blended. The frosting should be smooth and creamy.
Frost the cake when it is completely cool. Store the cake covered, in the refrigerator. (Stick toothpicks in the cake and gently cover it with foil.) Take out the cake several minutes before serving, for best taste. The cake will stay fresh in the refrigerator for three days if cut, or a week if left whole.
You can freeze carrot cake only if it is unfrosted. Wrap it in plastic, then foil, before freezing. Thaw it in the refrigerator.
To add nutsIf you wish to add nuts, decorate the outside of the frosted carrot cake with chopped walnuts or stir them into the batter before baking.Storing unfrosted carrot cakeCarrot cake should be refrigerated, even if unfrosted, since it contains carrots. Cover the unfrosted cake with plastic wrap and refrigerate it for up to a week.For a 10x15 panFor a 10x15 large glass pan, double the cake ingredients and use 1.5 times the frosting. Add 10-13 minutes to the baking time listed.For a three-layer carrot cakeFor a 9-inch round, three-layer cake, double the cake and frosting ingredients. Check the cake layers for doneness after 20 minutes.Recipe adapted from Mel's Kitchen Cafe.