1. In a saucepan, heat the oil over medium heat.
2. Sauté the onions and celery in the oil.
3. Pour in the broth and milk.
4. Add the ham, herbs, and seasoning. Bring to a boil.
5. Cover and simmer for 30 minutes over low heat.
6. Add the remaining ingredients.
7. Cook until the vegetables are tender and the ham easily falls apart.
8. Remove the ham or sausage from the saucepan and let it cool.
9. Slice it into thin strips and incorporate it into the soup.