Preheat oven to 350°F (180°C). Grease two 9-inch cake pans and put a parchment round in the bottom of each pan. Set aside.
In a medium mixing bowl, combine flour, baking powder, baking soda, ground cinnamon, nutmeg, allspice, and salt.
2 ½ cups all-purpose flour, 2 teaspoons baking powder, 1 teaspoon baking soda, 2 teaspoons ground cinnamon, ¼ teaspoon ground nutmeg, ¼ teaspoon allspice, 1 teaspoon salt
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.
4 large eggs, ½ cup unsweetened applesauce, ½ cup canola, 1 teaspoon vanilla extract, 1 cup brown sugar, ½ cup granulated sugar
Add dry ingredients to the wet ingredients and mix until just combined.
Fold in grated carrots, crushed pineapple, and walnuts.
2 cups grated carrot, 1 cup crushed canned pineapple, ½ cup chopped walnuts
Divide the batter between the cake pans and bake for 35-40 minutes, or until a skewer 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 powdered sugar in the bowl of a stand mixer fitted with the paddle attachment at medium speed until light and fluffy, about 2-3 minutes.
½ cup unsalted butter, 2 cups powdered sugar
Add vanilla extract and cream cheese and beat until blended. If the frosting is a little bit too soft, freeze it for a few minutes.
1 teaspoon vanilla extract, 16 oz block of cream cheese
Frost the cooled cake and decorate with chopped nuts.