Preheat the oven to 350 degrees. In a medium bowl combine the flour, baking powder, salt, cinnamon, nutmeg, and ginger. Stir until evenly combined.
Peel the carrot and use a large holed cheese grater to shred it into a separate bowl. Add the egg yolk, brown sugar, vanilla, and vegetable oil to the shredded carrot. Whisk these ingredients together with a fork until evenly combined.
Pour the bowl of dry ingredients into the bowl of wet ingredients and stir just until evenly combined.
Coat two small (6oz.) ramekins with non-stick spray, then divide the batter between the two. Place the ramekins on a baking sheet to make them easier to transport in and out of the oven. Bake the cakes for 30 minutes, or until a toothpick inserted in the center comes out clean.
While the cakes are baking, place the cream cheese, butter, and vanilla extract in a bowl and allow them to come to room temperature. Once room temperature and soft, whisk with a fork until smooth. Stir in the powdered sugar a little at a time until it is all incorporated.
Allow the cakes to cool before spreading the cream cheese frosting over top (otherwise it will melt and slide off). Enjoy!