For the Cake:

3 eggs
2/3 cup granulated sugar
2 tablespoons vegetable oil
1 teaspoon vanilla extract
1/2 teaspoon salt
1 teaspoon baking powder
1 teaspoon ginger
1/4 teaspoon nutmeg
2 teaspoons cinnamon
3/4 cup flour
2 cups shredded carrots (about 2 medium carrots)
Powdered sugar, to aid in rolling

For the Filling:

6 ounces cream cheese, room temperature
4 tablespoons butter, room temperature
2 cups powdered sugar
1 teaspoon vanilla extract
Powdered sugar, for dusting