Preheat oven to 350°F and lightly grease a large oven-safe ramekin.
In a medium bowl, whisk together the flour, brown sugar, granulated sugar, baking powder, cinnamon, nutmeg, and salt.
Add the oil, Greek yogurt, milk, and vanilla extract. Stir until just combined.
Fold in the grated carrots.
Pour batter into the prepared ramekin.
Bake for 22–28 minutes, or until the center is set and a toothpick inserted comes out with a few moist crumbs.
Allow the cake to cool for 15–20 minutes before frosting.
To make the frosting, beat together the cream cheese, butter, powdered sugar, and vanilla until smooth and creamy.
Spread frosting over the cooled mug cake and enjoy.