Preheat your oven to 350 degrees. Spray a 9x13 inch baking pan with non-stick cooking spray and line with parchment paper. Set aside
In a large bowl using a handheld mixer, beat the eggs, vegetable oil, apple sauce sugars, and vanilla until combined
Slowly mix in the flour, baking powder, baking soda, cinnamon, salt, and nutmeg until just combined
Fold in the shredded zucchini with a spatula until combined. Pour the mixture into the prepared baking pan and bake in preheated oven for 35-40 minutes or until a toothpick inserted in the center comes out clean
While the cake is baking, mix all the icing ingredients together until fully combined. Spread the icing on the completely cooled cake and sprinkle with cinnamon if desired