Preheat the oven to 350°F (175°C). Grease and flour two 9-inch round cake pans.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, nutmeg, and ginger until well combined.
In a large bowl, beat the granulated sugar, brown sugar, and eggs with an electric mixer on medium speed until light and fluffy, about 3 minutes.
Add the vegetable oil, applesauce, and vanilla extract to the sugar and egg mixture. Beat until fully combined.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Do not overmix.
Fold in the grated carrots, chopped walnuts, and drained crushed pineapple gently with a spatula until evenly distributed.
Divide the batter evenly between the prepared cake pans and smooth the tops with a spatula.
Bake 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 them out onto wire racks to cool completely.
While the cakes are cooling, prepare the cream cheese frosting. In a medium bowl, beat the softened cream cheese and butter together until smooth and creamy.
Gradually add the powdered sugar and vanilla extract, beating on low speed until combined. Increase speed to medium and beat until light and fluffy.
Once the cakes are completely cool, place one cake layer on a serving plate. Spread a generous layer of cream cheese frosting over the top.
Place the second cake layer on top and frost the top and sides of the cake with the remaining frosting.
Optional: Garnish with additional chopped walnuts or carrot decorations if desired.
Slice and serve. Store any leftovers in an airtight container in the refrigerator.