Preheat your oven to 350°F. Grease two 9-inch round cake pans and line the bottoms with parchment paper so the cakes release easily later.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger, and nutmeg until everything is evenly mixed.
In a large bowl, whisk together the brown sugar, granulated sugar, and vegetable oil until smooth. Add the eggs and whisk until fully combined. Then mix in the applesauce and vanilla.
Add the dry ingredients into the wet mixture. Stir gently until just combined. Do not overmix. Fold in the grated carrots and chopped nuts.
Divide the batter evenly between the prepared pans and smooth the tops.
Bake for 30 to 35 minutes, until the tops spring back lightly when touched and a toothpick inserted into the center comes out clean.
Let the cakes cool in the pans for about 10 minutes. Then run a knife around the edges, carefully turn them out, and place on a rack to cool completely.
If you want a flat layered cake, trim the top of one layer with a serrated knife.
Spread cream cheese frosting over the flat layer, place the second cake on top, and frost the top and sides. Add extra nuts on top if you like.
Slice and serve. You can also keep them as single-layer cakes if you prefer something simpler.