Preheat the oven to 350°F.
In a medium-sized mixing bowl, sift together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt. Set aside.
In 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 9×13-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 it to cool completely.
In a large mixing bowl, use a hand mixer on medium speed to beat the softened butter until the consistency is smooth. Add the vegan cream cheese and vanilla bean paste, then continue mixing for 30 seconds, just enough to incorporate with the butter. Switch the mixer speed to low, then begin gradually adding the powdered sugar, 1 cup at a time, until all is combined and the consistency of the frosting is thick but still spreadable. Store in the refrigerator until ready to use.
Once the cake has cooled completely, use a small spatula to spread the cream cheese frosting evenly over the top. Start in the center and work your way out to cover the entire surface, making sure to smooth it out for an even layer.