Preheat the oven to 350°F (175°C). Grease and flour two 9-inch round cake pans.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, ground cinnamon, ground nutmeg, and ground ginger until well combined.
In a large bowl, beat the granulated sugar, light brown sugar, and eggs together until light and fluffy, about 3 minutes.
Add the vegetable oil, unsweetened applesauce, and 2 teaspoons vanilla extract to the egg mixture and mix until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined. Do not overmix.
Fold in the grated carrots, chopped walnuts, and drained crushed pineapple until evenly distributed.
Divide the batter evenly between the prepared cake pans and smooth the tops with a spatula.
Bake in the preheated oven for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cakes from the oven and allow them to cool in the pans for 10 minutes. Then turn the cakes out onto wire racks to cool completely.
While the cakes cool, prepare the cream cheese frosting. In a large bowl, beat the softened cream cheese and unsalted butter together until smooth and creamy.
Gradually add the powdered sugar, one cup at a time, beating well after each addition.
Add 1 teaspoon vanilla extract and beat until the frosting is light and fluffy.
Once the cakes are completely cool, place one cake layer on a serving plate and spread a layer of frosting evenly over the top.
Place the second cake layer on top and spread the remaining frosting over the top and sides of the cake.
Refrigerate the cake for at least 30 minutes before serving to allow the frosting to set.