In a microwave-safe mug, melt the 1.5 tablespoons of butter.
Add the 2 tablespoons of brown sugar to the melted butter.
Add the quarter teaspoon of vanilla, a dash of cinnamon, and the quarter teaspoon of baking powder.
Whisk the mixture together thoroughly with a fork.
Add the quarter cup of all-purpose flour and the 3 tablespoons of milk, and whisk again with a fork until just combined.
Finally, fold in the 2 tablespoons of shredded carrots. Mix again.
Place the mug into the microwave and cook in 30-second increments until done. The total cooking time should be between 1:30 and 2:30, depending on the power of your microwave.