Preheat oven to 350°F (177°C). Spray two 9-inch round cake pans well with non stick cooking spray (you can also line the bottom of each pan with parchment paper for easier removal) and set aside.
In a large mixing bowl, whisk together the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt until well combined. Set aside.
In a separate large mixing bowl, whisk together the oil, eggs, brown sugar, granulated sugar, applesauce, and vanilla extract until fully combined. Add the grated carrots into the wet ingredients and mix until well combined.
Pour the wet ingredients into the dry ingredients and mix with a whisk or rubber spatula until just combined, making sure not to over mix the batter.
Pour the cake batter evenly between both prepared cake pans. Bake at 350°F for 30-35 minutes or until the tops of the cakes are set and a toothpick inserted into the center of each one comes out clean. Remove from the oven, transfer to a wire rack, and allow to cool in the pans for about 20-25 minutes. Once the cakes have cooled, remove from the pans and return the cakes to the wire rack to finish cooling.