Preheat the oven to 350 degrees F. Mist a 9x13-inch dish with nonstick cooking spray and set aside.
In a large bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and other spices; set aside. In a medium bowl, whisk together the oil, applesauce, brown sugar, granulated sugar, eggs, and vanilla extract. Pour the wet ingredients into the dry ingredients and mix until combined. Fold in the grated carrots, pineapple, and walnuts.
Spread batter into the prepared pan. Bake for 45-55 minutes. The cake is done when a toothpick inserted in the center comes out clean. Allow to cool completely.
To make the frosting, combine all frosting ingredients in a large bowl or bowl of a stand mixer. Mix for 2-3 minutes on medium-high or until smooth. Spread the frosting on the cooled cake. Refrigerate for 30 minutes or up to 2-3 days before serving.