Bring 3 cups water to a boil in a medium saucepan; add rice. Cover, reduce heat, and simmer 12 minutes. Remove from heat; let stand.
Bring 2 cups water and broth to a boil in a large saucepan. Stir in miso, soy sauce, and sugar. Add mushrooms, ginger, and garlic; reduce heat, and simmer for 2 minutes. Stir in carrot and onions; cook 2 minutes.
Whisk egg whites and egg in a small bowl. Gently pour egg mixture into broth mixture (do not stir); cook for 1 minute. Remove from heat; gently stir.
Spoon 1 cup of rice into each of 6 large soup bowls. Ladle 1 cup of the broth mixture evenly over rice.
Note: Sushi rice, a sticky rice that's available in many large supermarkets, holds together well when topped with vegetables and broth. But any rice will work in this dish.