2. Using a hand or stand mixer, mix the eggs, oil, applesauce, and vanilla extract on low speed for 30 seconds. Beat in both sugars on medium speed until well combined, 2 minutes. Add the flour, baking powder, cinnamon, salt, and nutmeg. Mix the batter until the dry ingredients are fully incorporated. Stir in the carrots on low speed until they are evenly distributed within the batter. Pour the batter into the baking dish and spread it evenly with a spatula.