Preheat oven to 350 degrees. Spray a 9x13" baking pan with non-stick cooking spray.
Beat eggs, vegetable oil, applesauce, white sugar, brown sugar, and vanilla together in a large bowl. Add in flour, baking powder, baking soda, cinnamon, salt, and nutmeg and mix well. Fold in zucchini and stir until completely mixed in. Pour batter into prepared 9x13" pan.
Bake for 35-40 minutes or until toothpick inserted into the center comes out clean.
While the cake cools, mix together frosting ingredients and spread over the top of the cake. If desired, sprinkle the top of the cake with cinnamon and sugar.