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Preheat the oven to 350 degrees and line two muffin cups with paper liners.
Peel the carrot and use a cheese grater to grate the carrot.
In a medium bowl, use a fork or rubber spatula to mix together the egg yolk, brown sugar, vanilla, and oil until combined. Mix in the grated carrot.
Fold in the flour, baking powder, salt, cinnamon, and nutmeg until just combined.
Divide batter among the two muffin cups.
Bake in the preheated oven for about 25 minutes, until the carrot cakes are golden brown and when you press the tops they slowly bounce back. You can also use a toothpick to test - there should be moist crumbs only, no wet batter.
While the carrot cake cools, mix up the frosting in a small bowl. Stir together the softened butter and cream cheese then stir in the vanilla and powdered sugar.
Frost the carrot cakes either using a piping bag or just spooning on top.
Enjoy!