Preheat your oven to 350 F. Spray a 9x13-inch baking pan with non-stick cooking spray, and line it 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, and salt just until combined.
Fold in the shredded zucchini with a spatula. Pour the mixture into the prepared baking pan and bake 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 until fully combined. Spread the icing over the completely cooled cake, and sprinkle with cinnamon if desired.