Preheat the oven to 350°F.
Into a medium-size mixing bowl, sift together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt. Set aside.
Into a large mixing bowl, stir together the sugar and applesauce. Add the oil and stir to incorporate into the mixture.
Gradually mix in the sifted dry ingredients.
Fold in the carrots and nuts, stirring just enough to distribute throughout the batter. Pour the batter into a prepared 9x13-inch metal baking dish.
Bake for 40-45 minutes, until a toothpick inserted into the cake comes out clean. Set the cake on a cooling rack and allow to cool completely before spreading with vegan cream cheese frosting.