Bring the water and ginger to a boil in a 2-quart saucepan. Add carrot, cover, and remove from heat. Let sit for 1 minute.
Whisk together shiro miso and 1/2 cup of the hot water in a small bowl until smooth, then whisk into the hot water. Remove and discard the ginger. Add tofu and scallions, divide among bowls, and serve immediately.