Sauté celery, onion, and green pepper in butter in a 4-quart Dutch oven until tender. Stir in boiling water, tomato, and salt, mixing well. Simmer, uncovered, 15 minutes. Carefully break eggs into hot soup. (Do not stir.) Add salt and pepper.
Sprinkle cheese over egg and soup mixture. Cover; simmer 5 minutes. Serve immediately over rice in individual soup bowls.