Preheat the oven to 350°F (175°C). Grease and flour a 9×13-inch baking pan.
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 and well blended.
Add the eggs one at a time to the sugar and oil mixture, beating well after each addition.
Stir in the 2 teaspoons of vanilla extract.
Gradually add the dry ingredients to the wet ingredients, mixing just until combined.
Fold in the grated carrots, shredded coconut, and chopped walnuts until evenly distributed.
Pour the batter into the prepared baking pan and spread it evenly.
Bake for 38 to 42 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 unsalted butter together until smooth and creamy.
Gradually add the powdered sugar, one cup at a time, beating well after each addition.
Stir in 1 teaspoon of vanilla extract and beat until the frosting is light and fluffy.
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.