To make cake, sift the whole wheat pastry flour, sugar, baking soda, baking powder, cinnamon and salt into a bowl.
In another bowl mix the oil, applesauce, grated carrots, and vanilla.
Add the flour mixture to the oil-applesauce carrot mixture and stir everything to blend evenly.
Add walnuts and mix.
Grease and flour 2 9-inch cake pans. Line the bottoms with parchment paper, if posibble because this makes it far easier to unmold the cake. Oil the parchment paper as well.
Now divide the batter evenly between the two pans
and bake at 350 degrees for 35 minutes or until a toothpick inserted in the center comes out clean.
Place the cake pans on a rack for 15 minutes. Then unmold the cakes gently and allow them to cool thoroughly on the rack. Frost when completely cooled.
To make frosting, beat together all of the frosting ingredients in a bowl until very smooth.
To frost, place one of the cooled cakes on a cake stand or plate. Smear the top with some frosting.
Carefully place the other cake on top of the first one. Cover the cake with the remaining icing using a spatula.
Make the Frosting:
Beat together the cream cheese, sugar and vanilla extract until really smooth
Place one of the cake layers on a cake stand or plate. Smear the top with icing.
Place the second layer on top of the first and spread the remaining icing on top and over the sides of the cake using a spatula or ladle.