Preheat oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a large bowl, whisk together the flour, baking powder, baking soda, cinnamon, nutmeg, cloves, and salt.
In another bowl, whisk together the eggs, granulated sugar, brown sugar, oil, applesauce, and vanilla extract until smooth.
Gradually stir the dry ingredients into the wet mixture, just until combined. Fold in the grated carrots and optional nuts.
Pour the batter into the prepared pan and spread evenly. Bake for 35–40 minutes or until a toothpick inserted in the center comes out clean.
Let the cake cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.
For the frosting, beat together the cream cheese, butter, powdered sugar, and vanilla until smooth. Spread over the cooled cake before serving.