Preheat the oven to 350°F (175°C) and grease a 9x13-inch baking pan, lining it with parchment paper.

In a medium bowl, mix all-purpose flour, baking powder, baking soda, salt, cinnamon, and nutmeg.

In a larger bowl, beat the eggs, brown sugar, granulated sugar, vegetable oil, applesauce, and vanilla until smooth.

Gradually add the dry mixture to the wet ingredients, then fold in grated carrots and optional nuts.

Pour the batter into the prepared pan and bake for 25–30 minutes. Check for doneness with a toothpick.

Cool the bars in the pan on a wire rack for about 1 hour.

For the frosting, beat together softened cream cheese and butter until smooth, then add powdered sugar and vanilla.

Once cooled, frost the bars with the cream cheese frosting, cut into squares, and serve.