Preheat the oven to 350°F (180°C). Spray two 9-inch round cake pans with non-stick cooking spray and line the bottoms with circles of parchment paper. Set aside.

In a large mixing bowl, whisk together the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt until well combined. Set this bowl aside.

In another large mixing bowl, whisk the oil, eggs, brown sugar, granulated sugar, applesauce, and vanilla extract until everything is mixed well. Add the grated carrots and mix again until combined.

Pour the wet ingredients into the bowl with the dry ingredients. Mix gently with a whisk or rubber spatula until just combined.

Divide the batter evenly between the two prepared cake pans. Bake for 30 to 35 minutes, or until the tops are set and a toothpick inserted into the center comes out clean.

Once baked, remove from the oven and let them cool in the pans for 20 to 25 minutes. After cooling, gently remove the cakes from the pans and place them on a wire rack to cool completely.

To prepare the cream cheese frosting, beat the softened cream cheese in a mixing bowl until smooth. Add the softened butter and mix for about 30 seconds until smooth and creamy.

Gradually mix in the powdered sugar and vanilla extract, and continue mixing until fully combined. Scrape down the sides of the bowl if needed.

To assemble the cake, level the tops of the cakes. Place one layer on a cake stand, and spread a little over ½ cup of frosting on top evenly. Place the second layer on top and use the remaining frosting to frost the top and sides of the cake.

Optionally, you can top it with pecans or other favorite toppings.