Heat the vegetable oil in a 7-quart Dutch oven over medium-high heat. Add the onion, bell pepper, celery, salt, and pepper. Cook, stirring frequently, until the onions and celery are semi-translucent and the bell peppers are tender, 6 to 8 minutes. Add the garlic and cook for 1 to 2 minutes, stirring constantly. Add the pickled pork, bay leaves, thyme, cayenne pepper, water, and beans to the pot and increase the heat to high. Cook, stirring frequently, until the mixture comes to a boil, approximately 6 to 8 minutes.
Decrease the heat to maintain a simmer, cover, and cook for 1 1/2 hours, stirring every 30 minutes. Uncover, increase the heat slightly to maintain a steady simmer and continue to cook until the beans are tender and the sauce is thickened to your liking, another 30 to 40 minutes. If you prefer an even creamier texture, mash some of the beans with a potato masher or pulse it a few times with an immersion blender.
Prepare rice during the last 30 minutes of cooking the beans. Place the water into an electric kettle and bring to a boil. While the water is coming to a boil, place the butter into a 3-quart saucepan set over medium heat. Once the butter begins to bubble, add the rice and stir to combine. Add the salt and cook for 2 to 3 minutes, stirring frequently. Carefully pour the water over the rice and stir to combine. Decrease the heat to the lowest setting, cover, and cook for 15 minutes. Kill the heat and let the rice sit, with the lid on, for 5 more minutes. Remove the lid and fluff the rice with a fork. Serve the beans over the rice.