Pat the pork chops dry with paper towels. In a small bowl, combine salt, pepper, and paprika. Season both sides of the pork chops generously.
Heat a large cast iron skillet or Dutch oven over medium-high heat and add the vegetable oil. Once hot, sear the pork chops for 3-4 minutes on each side until golden brown. Work in batches if necessary to prevent overcrowding.
Remove the seared pork chops and set aside on a plate. In the same skillet, reduce heat to medium and add the sliced onions. Sauté for 2-3 minutes until they begin to soften.
Add the minced garlic and cook for an additional 30 seconds until fragrant. Stir occasionally to scrape up any browned bits from the bottom of the pan.
Add the quartered potatoes and carrots to the skillet, arranging them in a single layer as much as possible. Season with a pinch of salt and pepper.
Pour in the chicken broth and sprinkle the thyme leaves over the vegetables.
Nestle the seared pork chops on top of the vegetable layer. Cover the skillet with a tight-fitting lid and reduce heat to low. Simmer for about 25 minutes or until the potatoes begin to soften.
Add the green beans on top, replace the lid, and continue cooking for another 10-15 minutes until all vegetables are tender and the pork chops reach an internal temperature of 145°F (63°C).
Transfer the pork chops and vegetables to a serving platter. If you prefer a thicker sauce, whisk the flour with 2 tablespoons of cold water and stir into the remaining liquid in the skillet. Cook for 2-3 minutes until thickened.
Add the butter and stir until melted for a silky finish. Pour the sauce over the pork chops and vegetables before serving.