Preheat oven to 350 degrees F. Spray a 15 x 10 in., rimmed baking sheet with cooking spray.
In a large bowl, beat eggs & sugar well with electric mixer, until mixture is light in color. Add oil or oil & applesauce & mix in well with mixer. Then, add in pumpkin puree, & mix well with mixer.
In a separate bowl, combine all dry ingredients with a whisk, making sure ingredients are well mixed. Add dry mixture to wet mixture, a cup at a time, mixing in between, gently with mixer or a wooden spoon. Don't overmix, at this point. Just make sure there isn't any flour mixture left showing.
Spread batter in prepared pan. Bake 25-30 minutes.
Cool in pan, on rack, away from the stove. After it is totally cooled, prepare frosting.
Frost with cream cheese frosting: Beat softened cream cheese & butter & vanilla, until mixed well. Add powdered sugar in 1/2 cup amounts, at a time, until all of it is incorporated. Spread on cooled cake. Cut in bars.