Sauté onion in 2 tbsp. butter in a heavy saucepan until tender, about 3 minutes. Add cabbage, beans, carrots, potatoes, broth and salt and then cover and simmer until carrots and potatoes are tender, about 20 minutes.
Meanwhile, melt 1/4-cup butter in a 4-quart Dutch oven. Blend in flour, paprika and pepper. Remove from heat; stir in milk. Heat to boiling, stirring constantly. Boil and stir 1 minute.
Remove from heat; stir in cheese until melted. If necessary, return to low heat to finish melting cheese, without boiling. Add vegetables with liquid to cheese soup base. Heat to serving temperature.