Preheat oven to 350°F. Lightly coat an 8-inch cake pan with nonstick spray and line with a parchment round.
In a large bowl, beat together eggs, applesauce, and sugar, then mix in carrots.
In another bowl, sift together flours, baking powder, cinnamon, nutmeg, and salt, then stir into carrot mixture until completely mixed in. The batter will be thick.
Scoop the batter into the prepared pan and spread to the edges in an even layer.
Bake for 18-22 minutes, until a toothpick inserted in the center comes out clean. Let cool in the pan 5-10 minutes, then turn out onto a wire rack to cool completely before frosting.
To frost, beat together neufchatel cheese, powdered sugar, vanilla, and lemon zest until smooth. Frost cake and top with walnuts, if desired.