Preheat oven to 350°F (175°C) and grease two 4-ounce ramekins with butter or cooking spray.
In a medium bowl, whisk together the flour, sugar, baking soda, baking powder, cinnamon, ginger, and salt.
Add the vegetable oil, egg yolk, and vanilla extract to the dry ingredients and stir until just combined.
Fold in the finely grated carrots and chopped walnuts until evenly distributed.
Divide the batter equally between the two prepared ramekins.
Bake for 22 to 25 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and allow the cakes to cool completely on a wire rack.
In a small bowl, beat the cream cheese, butter, and powdered sugar until smooth and creamy.
Spread the frosting over the top of the cooled cakes before serving.