In a large, heavy pot, heat the bacon drippings over medium heat.
Add the onions, bell peppers and celery and cook, stirring, until they are lightly golden, about 10 minutes.
Add the garlic and cook for three minutes. Add the ham, sausage and salt pork. Cook for five minutes.
Add the beans and add enough shrimp or chicken broth to cover. Season with salt, cayenne and black pepper.
Add the thyme, basil and bay leaves. Bring to a boil, then reduce heat to medium-low. Cook, partially covered, for about two hours, stirring occasionally, or until the beans are tender.
Add more water or broth if the mixture becomes dry. If you like the beans to be creamy, puree about one-third of the beans in a food processor, in batches, and return to the pot.
Cook for another 30 minutes. Remove and discard the bay leaves. Serve hot, garnished with green onions and parsley, over rice.