In large pan saute onion and 1/4 cup butter until tender. Add potatoes, carrots, celery, 1 teaspoon salt and 2 cups water. Simmer 20 to 25 minutes. In medium saucepan melt 1/4 cup butter; add 1 tablespoon salt and 1/4 teaspoon pepper.
Cook until bubbly. Gradually add 4 cups milk and bouillon cubes. Cook until mixture thickens. Add to vegetable mixture and simmer until warm. Add cheese and blend in.