Preheat the oven to 350°F (180°C) and spray two 9-inch cake pans* with non-stick spray and line the bottoms with circles of parchment paper.
Sift the all purpose flour into a mixing bowl and add the sugar, baking soda, baking powder, salt, cinnamon and nutmeg and mix together.
Add the eggs, grated carrot, oil, applesauce, vanilla extract and white vinegar and mix together into a thick batter. If it seems like it won't come together, then just leave the bowl for a couple of minutes (the carrots will release more moisture into the batter) and then return to it and mix it in. As long as you used fresh carrot that you grated yourself it will mix into a batter.
Add the chopped walnuts and fold them in.
Divide the batter between your two prepared cake pans.
Bake for 30 minutes or until a toothpick inserted into the center comes out clean.
Leave to cool for a few minutes and then remove the cakes from the cake pans and transfer them to a wire cooling rack to cool completely before frosting with cream cheese frosting.