To Make the Carrot Cake:
Preheat the oven to 350°F (180°C). Spray two 9-inch round cake pans with non-stick cooking spray and line the bottom of each pan with a parchment paper circle. 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 to 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. Be careful not to overmix.
Divide the cake batter evenly between the prepared cake pans. Bake for 30 to 35 minutes or until the tops of the cakes are set and a toothpick inserted into the center comes out clean.
Remove the cakes from the oven and place them on a wire rack to cool in the pans for 20 to 25 minutes. After cooling, remove the cakes from the pans and allow them to cool completely on the wire rack.
To Make the Cream Cheese Frosting:
In the bowl of a stand mixer fitted with the paddle attachment, or using a hand-held mixer in a large mixing bowl, beat the cream cheese until smooth. Add the butter and mix for about 30 seconds to 1 minute until well combined and smooth.
Add the powdered sugar and vanilla extract and continue mixing until fully combined, stopping occasionally to scrape down the sides of the bowl.
To Assemble the Cake:
Level the tops of the cooled cakes using a knife or cake leveler.
Place one cake layer on a cake stand, spread a little over 1/2 cup of frosting evenly over the top. Place the second cake layer on top and use the remaining frosting to cover the top and sides of the cake.
Optionally, garnish with pecans or other toppings of choice.