Preheat oven to 350°F (175°C). Grease and line a 9x13-inch baking pan.
In a medium bowl, whisk together flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
In a large bowl, beat eggs, brown sugar, granulated sugar, oil, applesauce, and vanilla until smooth.
Add dry ingredients to wet ingredients and mix just until combined. Fold in grated carrots and nuts if using.
Pour batter into prepared pan. Bake for 25–30 minutes or until a toothpick inserted comes out clean. Let cool completely.
Beat softened cream cheese and butter until creamy. Add powdered sugar and vanilla, and beat until smooth and fluffy.
Spread frosting over cooled bars. Cut into squares and serve.
