In a microwave-safe mug (at least 12 ounces), whisk together the flour, granulated sugar, baking powder, cinnamon, nutmeg, and salt until no lumps remain.
Add the milk, oil, and vanilla extract to the dry ingredients. Stir with a fork or small whisk until the batter is smooth and well combined, about 20 strokes.
Fold in the grated carrot, making sure it's evenly distributed throughout the batter. The mixture should look thick but pourable.
Microwave on high power for 90 seconds. The cake will rise and look set on top but still slightly moist. Let it rest in the microwave for 30 seconds to finish cooking with residual heat.
While the cake cools slightly, mix the softened cream cheese, powdered sugar, and milk in a small bowl until smooth and spreadable. Adjust consistency with more milk if needed.
Spread or dollop the cream cheese frosting over the warm cake. Let it melt slightly into the surface, creating a luscious glaze. Serve immediately with a spoon.