Preheat oven to 350°F (175°C). Grease and flour two 9-inch round cake pans.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In a large bowl, beat the granulated sugar, brown sugar, and eggs until smooth and creamy.
Add the vegetable oil, applesauce, and vanilla extract to the sugar and egg mixture and mix until combined.
Gradually add the dry ingredients to the wet ingredients, mixing just until incorporated.
Fold in the grated carrots and chopped walnuts until evenly distributed.
Divide the batter evenly between the prepared cake pans and smooth the tops.
Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
Remove cakes from the oven and allow them to cool in the pans for 10 minutes.
Turn the cakes out onto wire racks and cool completely before frosting.
To make the cream cheese frosting, beat the softened cream cheese and butter together until smooth and creamy.
Gradually add the powdered sugar, one cup at a time, beating well after each addition.
Add the vanilla extract and beat until the frosting is light and fluffy.
Place one cake layer on a serving plate and spread a layer of frosting on top.
Place the second cake layer on top and spread frosting evenly over the top and sides of the cake.
Refrigerate the cake for at least 30 minutes before serving to allow the frosting to set.