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, ground cinnamon, ground nutmeg, and salt until well combined.
In a large bowl, beat the granulated sugar, light brown sugar, and vegetable oil until smooth.
Add the eggs one at a time to the sugar and oil mixture, beating well after each addition.
Stir in 2 teaspoons vanilla extract.
Gradually add the dry flour mixture to the wet ingredients, mixing just until combined.
Fold in the grated carrots, chopped walnuts, drained crushed pineapple, and shredded coconut until evenly distributed.
Divide the batter evenly between the two prepared cake pans and smooth the tops.
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.
Turn the cakes out onto wire racks and cool completely before frosting.
To make the frosting, 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 continue to beat until the frosting is light and fluffy.
Place one cake layer on a serving plate and spread a thick layer of cream cheese frosting over the top.
Place the second cake layer on top and frost the top and sides evenly with the remaining frosting.
Refrigerate the cake for at least 30 minutes before serving to allow the frosting to set.