Remove meat from ham bone, reserve bone. Cut meat into small pieces. Melt butter in a large soup pot or dutch oven over low heat. Add ham meat (if using), onions, celery and leeks. Cook for 20 minutes or until vegetables are soft. Add broth, water and ham bone (if using) and bring to a boil. Add cabbage. Reduce heat and simmer for 30 minutes. Add 1 pound carrots, 1 pound turnips and potatoes. Simmer one hour.
Tie together with twine or put in a cheesecloth bag the parsley, thyme and bay leaf to make a bouquet garni.
While soup is cooking, drain beans and transfer them to a medium saucepan with enough water to cover beans plus about 1 1/2 inches. Add two chopped turnips, 2 chopped carrots, garlic and the bouquet garni (parsley, thyme and bay leaf). Simmer for 1 1/2 hours or until beans are tender. Drain beans and discard bouquet garni. Add beans to soup and bring to a boil. Season with salt and pepper.
To serve place two toast halves in each soup bowl, top with crumbled cheese and ladle soup over toast.