Heat oven to 325 degrees. Butter and lightly flour 2 13-by-9 inch sheet cake pans (or 4 9-by-9 inch square cake pans).
Using large stand mixer (like Kitchen-Aid brand), combine all dry ingredients until thoroughly mixed. Add carrots, oil (or oil and applesauce) and eggs. Beat on low until mixed, then increase mixer speed to medium and beat for two minutes. Spread batter in pans.
Bake at 325 degrees until the cake centers spring back when touched lightly--about 45 minutes for large pans, 35 to 40 minutes for small pans. Allow cakes to cool in the pan on a cooling rack before frosting with Cream Cheese icing.
Cream Cheese Icing: Allow 2 3-ounce packages of cream cheese and one (1/2 cup) stick of butter to come to room temperature. Place in mixer bowl, and beat on high speed until combined. Add 2 teaspoons vanilla and 4 cups powdered sugar. Beat at medium-high speed until light and fluffy--2 to 4 minutes. Ice tops of cakes only. Sprinkle frosting with chopped walnuts.
Serves/Yields: 12 - 15 servings (two large cakes or 4 small cakes).