Sauté green onion in butter in a large Dutch oven until tender; add flour, and stir well. Cook 1 minute, stirring constantly. Gradually add broth, half-and-half, milk, vermouth, and water; cook over low heat, stirring constantly, until slightly thickened and bubbly. Add salt, pepper, and hot sauce; stir well.
Bring to a boil. Reduce heat; simmer, uncovered, 30 minutes. Serve in warm soup bowls.