Preheat the oven to 350 degrees F.
Grease a 15x10x1 baking pan. Set aside.
In a medium bowl, whisk together the four, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a large bowl, beat together the sugar, oil, applesauce, eggs, and vanilla.
Gradually add the flour mixture, mixing after each addition and scraping down the sides of the bowl.
Stir in the carrots and nuts, if using, until well combined.
Spread the batter evenly into the prepared baking pan.
Bake for 20-25 minutes, or until a toothpick inserted in the center comes out clean.
Cool completely on a wire rack.