Preheat oven to 350°F (180°C). Spray two 9-inch cake pans with nonstick spray. Set aside.
In a medium bowl combine flour, baking powder, baking soda, ground cinnamon, nutmeg, allspice, and salt. Set aside.
In another large bowl whisk together eggs, applesauce, oil, vanilla extract, brown sugar, and granulated sugar. Mix with a whisk until no brown sugar lumps remain and the mixture is well combined.
Add dry ingredients to the wet ingredients and mix until just combined.
Fold in grated carrots, crushed pineapple, and walnuts.
Divide batter between the cake pans and bake for 35-40 minutes, or until a toothpick comes out clean.
Let the cakes cool completely in the pans set on a cooling rack.
To make the frosting, beat together softened butter and confectioners' sugar in the bowl of a stand mixer fitted with the paddle attachment at medium speed until light and fluffy, about 2-3 minutes.
Add vanilla extract and cream cheese and beat until blended. If the frosting is a little bit too soft, freeze for a few minutes.
Frost the cooled cake and decorate with chopped nuts.