Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan or line it with parchment paper.
In a large bowl, whisk together the all-purpose flour, baking powder, baking soda, salt, ground cinnamon, ground nutmeg, and ground ginger until well combined.
In a separate medium bowl, combine the granulated sugar, light brown sugar, applesauce, vegetable oil, and vanilla extract. Mix until smooth and fully incorporated.
Pour the wet ingredients into the dry ingredients and stir gently until just combined. Do not overmix.
Fold in the grated carrots, crushed pineapple, and chopped walnuts (if using) until evenly distributed.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 38 to 42 minutes, or until a toothpick inserted into the center of the cake comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 15 minutes. Then transfer the cake to a wire rack to cool completely.
While the cake cools, prepare the frosting. In a medium bowl, beat the softened cream cheese and unsalted butter together until smooth and creamy.
Gradually add the powdered sugar and vanilla extract, beating until the frosting is light and fluffy.
Once the cake is completely cooled, spread the cream cheese frosting evenly over the top of the cake.
Slice and serve. Store any leftovers covered in the refrigerator.