Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.
Beat eggs, vegetable oil, white sugar, and vanilla extract in a large bowl until thoroughly combined. Whisk flour, baking powder, baking soda, and salt in a separate bowl. Stir the flour mixture into the wet ingredients and mix in zucchini. Transfer batter to the prepared baking dish.
Bake in the preheated oven until the cake is lightly browned and a toothpick inserted into the center comes out with moist crumbs, about 40 minutes. Let cool completely.
Beat cream cheese and butter in a bowl until smooth; stir in applesauce and nutmeg. Mix in confectioners' sugar until the frosting is smooth and workable; spread on cooled zucchini cake.
