Preheat oven to 350. Spray 10 x 15 jelly roll pan with vegetable oil cooking spray or line with parchment paper.
Beat pumpkin, sugar, and oil until well combined. Add eggs, 1 at a time, and beat until blended, scraping down the bowl as necessary. Mix dry ingredients separately and add them one cup at a time, blending on low after each addition. Stir in applesauce and shredded apples by hand.
Pour batter in to prepared pan, gently spreading to smooth it out.
Bake at 350 for 25-30 minutes. Cool in pan on wire rack. After 15 minutes of cooling, cover with clean tea towel.
After cake is cooled, make icing: blend butter, cream cheese, and vanilla with mixer until fluffy. Slowly add powdered sugar and beat until smooth.
Spread icing on cake and serve directly out of the pan. Refrigerate any leftovers.