Place a 7 quart cast iron Dutch oven over medium-high heat. Add oil and heat until shimmering. Add onion, bell pepper and celery. Sprinkle with salt and pepper. Cook, stirring frequently, until onion and celery are soft and translucent, 6-8 minutes. Add garlic, pickled pork, bay leaves, dried thyme, cayenne pepper, hot sauce and beans. Add water. Turn up to high and heat until boiling, stirring occasionally, 6-8 minutes.
Reduce heat to maintain a strong simmer. Cover tightly and cook 90 minutes, stirring every 30 minutes. Remove the lid and cook, uncovered, 30-40 minutes longer, maintaining a strong simmer. If you want your sauce a bit thicker and more gravy-like, use a potato masher to crush some of the beans. Remove bay leaves.
Bring 3 cups water to a full rolling boil. Meanwhile, in a medium saucepan, melt butter over high heat. When foaming subsides and butter begins to brown, add rice and salt. Saute until rice turns opaque and slightly tan. Add boiling water all at once - be very careful! Cover tightly, reduce to simmer and cook 17-20 minutes, until fully cooked. If you prefer your rice dry and fluffy, let stand off heat, uncovered, 5 minutes before serving. Serve beans over rice.