Preheat your oven to 350°F (175°C) and grease and flour two 9-inch round cake pans.
In a medium bowl, whisk together flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In a large bowl, beat together the eggs, vegetable oil, granulated sugar, brown sugar, and vanilla extract until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
Fold in the grated carrots and chopped walnuts until evenly distributed.
Divide the batter evenly between the prepared cake pans.
Bake for 40-45 minutes or until a toothpick inserted into the center of the cakes comes out clean.
Allow the cakes to cool in the pans for about 10 minutes, then turn them out onto a wire rack to cool completely before frosting.