Preheat oven to 350 degrees F.
Grease 9x13" baking dish with cooking spray. Set aside.
In a large bowl with electric mixer, beat together the eggs, sugar, oil, applesauce, and pumpkin puree.
In a medium bowl, combine flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
Slowly add dry ingredients to wet ingredients, mixing in low until just combined.
Pour into prepared pan and bake in preheated oven for 28-32 minutes, until toothpick comes out of the middle clean.
Let cool completely.