Begin by placing the kidney beans in a large bowl. Cover the beans with 6 cups (1.5L) of cold water, stirring in 2 tablespoons (30g) of kosher salt until dissolved. Let the beans soak at room temperature for 8 to 16 hours. Once ready, drain and rinse them.
In a large Dutch oven, heat the vegetable oil or lard over medium-high heat. Once the oil shimmers, add the andouille sausage and cook, stirring occasionally, until the pieces are lightly browned, about 5 minutes. Toss in the onion, green bell pepper, and celery. Season with a pinch of salt, and continue cooking, stirring regularly, until the vegetables are soft and beginning to brown slightly around the edges, about 8 minutes.
Next, stir in the minced garlic and let it cook until fragrant, around 45 seconds. Follow this by adding the cayenne pepper, ground sage, and a generous amount of freshly ground black pepper (10 to 12 grinds). Stir everything together for about 30 seconds, allowing the spices to become aromatic.
Now, add the soaked and drained beans to the pot. Pour in enough water to cover the beans by about 2 inches (roughly 6 to 8 cups). If using, add the smoked ham hock and pickled pork. Toss in the thyme sprigs and bay leaves. Bring everything to a boil, then reduce the heat to a low simmer. Cover the pot and cook until the beans are tender, which will take between 1 1/2 to 2 1/2 hours, depending on the age of your beans.
Once the beans are fully tender, remove the lid and let the mixture continue simmering, stirring occasionally, for an additional 20 minutes, or until the liquid has thickened to a creamy consistency. If the mixture appears too dry before it becomes creamy, add more water, one cup at a time, and continue cooking. Repeat this process as necessary until the stew reaches the desired creaminess.
Before serving, discard the thyme stems and bay leaves. Adjust the seasoning with hot sauce, a splash of cider vinegar (if using), and additional salt and pepper to taste. For optimal texture, allow the stew to cool, then refrigerate overnight. Reheat the dish the next day, adding a little water to loosen the consistency if needed. Serve over freshly steamed white rice.