In a large Dutch oven, place the beans and 1 tablespoon of salt. Fill the Dutch oven with water to cover the beans. Put the lid on the Dutch oven and soak the beans overnight. (Or, for a quick soak, bring the pot of beans to a boil over medium-high heat for 2 minutes. Remove the pot from the heat, cover, and let stand for 1 hour.)

Whether they are soaked overnight or quick-soaked, drain and rinse the beans. Set them aside.

In a large Dutch oven over medium heat, heat 1 tablespoon of oil. Add the sausage and cook, stirring frequently, until browned and crisp, 5 to 7 minutes. Add the remaining 3 tablespoons of oil, along with the onion, bell pepper, and celery. Cook, stirring frequently, until the vegetables are soft and golden, 5 to 7 minutes. Stir in the garlic, thyme, black pepper, sage, cayenne, and the remaining 1 teaspoon of salt. Cook until fragrant, 30 seconds to 1 minute.

Stir in the chicken broth, ham hock, bay leaves, and drained beans. Add enough water to cover the beans by 2 inches, 2 to 4 cups. Bring to a boil, cover, and reduce the heat to low. Cook until the beans are tender, 1 hour and 15 minutes to 2 hours, checking the beans after 1 hour and 15 minutes for doneness.

Remove the lid and continue to cook over medium-low heat, stirring frequently, until the liquid is reduced and creamy, 25 to 30 minutes. Discard the bay leaves and stir in the vinegar. Taste for salt, pepper, cayenne, and vinegar.

Serve the beans over hot cooked rice with hot sauce and sliced green onion, if you like.