Preheat the oven to 350°F (175°C). Grease and flour a 9×13 inch baking pan.
In a medium bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, and salt until well combined.
In a large bowl, beat the eggs, granulated sugar, and brown sugar together until light and fluffy.
Add the vegetable oil, applesauce, and vanilla extract to the egg mixture and mix until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined.
Fold in the grated carrots, chopped walnuts, crushed pineapple, and shredded coconut until evenly distributed.
Pour the batter into the prepared baking pan and spread it evenly.
Bake in the preheated oven for 35-40 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool completely in the pan on a wire rack.
While the cake cools, 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 until the frosting is smooth and spreadable.
Once the cake is completely cooled, spread the cream cheese frosting evenly over the top.
Slice and serve. Store any leftovers covered in the refrigerator.