Line a 9 x 13 inch baking pan with parchment paper and set aside. For a three layer six-inch cake, spray sides of cake pans with baking spray and line pans with parchment paper. Preheat oven to 350 F.
In large mixing bowl, whisk to combine the flour, baking powder, baking soda, pumpkin pie spice, cinnamon, nutmeg, and salt. Set aside.
In a large mixing bowl, whisk to combine the sugars and eggs. Stir in the vanilla and pumpkin puree followed by the oil. Add the buttermilk alternatively with the dry ingredients, mixing just til combined. Stir in carrots and apple.
Pour the batter into the pan and spread it evenly in the pan. Bake until a toothpick comes out clean when inserted in the center of the cake, 28 to 38 minutes. Allow to cool completely before frosting.
In a large bowl or the bowl of a stand mixer fitted with the paddle attachment, mix the cream cheese and butter on medium speed for one minute until creamed and combined. Add the brown sugar and mix on medium for another minute. Mix in the vanilla then turn the mixer on low and gradually mix in the powdered sugar.
