According to a Wikipedia article, the ginger has a protease zingipain that curdles the milk. This allows it obtain a pudding like texture as it sits.
Heating the milk milk up in a microwave is a quick method that requires little clean up. However, do only heat it in 10 to 20 second increments. Or else, it would warm too quickly and you would need to wait for it to cool down before adding to the ginger juice.
The milk should be about 70°C in order for the enzymes in the ginger juice to work at optimal rate. If it is too hot, the enzymes would be denatured and be unable to curdle the milk.