In order to prepare the carrot cake, first preheat the oven to 350°F (180°C). Make sure to thoroughly spray two 9-inch round cake pans with non-stick cooking spray and place a circle of parchment paper at the bottom of each pan. Set these pans aside for now.
Combine the flour, baking powder, baking soda, cinnamon, ginger, nutmeg, and salt in a large mixing bowl, and whisk them together until they are well mixed. Set the bowl aside.
Combine the oil, eggs, brown sugar, granulated sugar, applesauce, and vanilla extract in a different large bowl, whisking until thoroughly blended. Mix in the grated carrots until fully incorporated.
Combine the wet ingredients with the dry ingredients by pouring them in and using a whisk or rubber spatula. Mix them together until they are just combined, taking care not to over mix the batter.
Divide the cake batter equally into both greased cake pans. Place in the oven and bake for approximately 30 to 35 minutes, or until the tops of the cakes are firm and a toothpick inserted into the center of each cake comes out without any batter clinging to it.
Take out of the oven, place on a wire rack, and let it cool in the pans for about 20 to 25 minutes. After the cakes have cooled down, take them out of the pans and put them back on the wire rack to continue cooling.
To prepare the cream cheese frosting: In a stand mixer with a paddle attachment, or in a large mixing bowl with a hand-held mixer, blend the cream cheese until it becomes smooth. Afterwards, add the butter and mix for approximately 30 seconds to 1 minute until the ingredients are fully combined and the mixture is smooth.
Include the powdered sugar and vanilla extract and keep blending until thoroughly mixed, pausing occasionally to scrape the sides of the bowl.
To put the cake together: Use a knife or cake leveler to make the tops of each cake even. Put one cake layer on a cake stand, add a little more than 1/2 cup of frosting on top, and spread it out evenly. Add the other cake layer on top and use the rest of the frosting to cover the top and sides of the cake. If desired, finish by adding pecans or other preferred toppings.