Preheat oven to 350°F and grease a 9×13-inch baking pan (or two 8″ round pans) with butter or the cooking oil of your choice.
Beat eggs, melted butter, applesauce, sugars, and vanilla together in a large bowl. This can be done by hand, with a hand mixer, or in the bowl of a stand mixer. Mix until light and fluffy.
Add the dry ingredients (flour, baking powder, baking soda, cinnamon, salt, and nutmeg) to the mixing bowl. Mix well until completely incorporated.
Gently stir in shredded zucchini until evenly distributed.
Pour batter into the prepared pan(s). Bake for 35-40 minutes until a toothpick inserted into the center comes out clean. Let cool before cutting.