Method
Preheat oven to 350°F (177°C). Grease and line two 9-inch cake pans.
Whisk flour, cinnamon, nutmeg, baking soda, and salt together.
In a separate bowl whisk together the oil, granulated sugar, and brown sugar.
Add eggs one at a time, mixing until smooth. Fold in grated carrots.
Fold dry ingredients into wet ingredients just until no dry flour remains.
Divide batter between pans and bake 25–30 minutes, or until a toothpick comes out clean.
Cool cakes in pans 10 minutes, then transfer to a rack and cool completely.
Make frosting: beat cream cheese and butter until smooth and creamy.
Add powdered sugar gradually, then mix in vanilla until smooth.
Frost cooled cake layers and assemble.
Notes
Flavor improves overnight. Store covered in the refrigerator up to 4 days.