Brown the pork: Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Season the pork with salt, pepper, and smoked paprika. Add the pork and brown on all sides, about 5–7 minutes.
Cook the aromatics: Add the diced onion and cook for 3–4 minutes until softened. Stir in the garlic and cook for another 30 seconds.
Add the broth: Pour in the chicken broth and add the thyme and bay leaf. Stir well, scraping up any browned bits from the bottom of the pot.
Simmer: Bring to a gentle boil, then reduce the heat to low. Cover and simmer for about 50–60 minutes, or until the pork is tender.
Add the beans: Stir in the drained butter beans. Simmer uncovered for another 10–15 minutes, allowing the beans to heat through and the broth to thicken slightly.
Adjust the seasoning: Remove the bay leaf and taste. Add more salt and black pepper if needed.
Serve: Spoon the pork and butter beans into bowls and serve hot with cornbread, rice, or crusty bread.