Pat the potatoes thoroughly dry. Heat 1½ tablespoons oil in the wok over medium-high heat. Add the potatoes, working in batches if needed to avoid overcrowding. Cook for 8–10 minutes per batch, turning occasionally and letting them sit between turns to develop golden edges.
Return all potatoes to the wok. Reduce heat to medium-low and carefully add ¼ cup water, keeping your hands away from the steam. Cover and cook for 5–7 minutes, until fork-tender. Add a small splash of water if needed. Uncover and let the water evaporate, then increase heat to medium-high and cook for another 2–3 minutes to brown again. Toss with ½ teaspoon Cajun seasoning. Transfer to a large plate and leave uncovered.
Divide the remaining 1½ tablespoons oil between two batches of green beans. Cook over medium-high heat for 5–7 minutes per batch, tossing occasionally and allowing the beans to rest against the hot surface between tosses. For softer beans, reduce heat, carefully add a splash of water, and cover briefly. Uncover and let the water evaporate. Transfer to a plate.
Add both meats to the wok over medium-high heat. Cook for 8–10 minutes, breaking them into small crumbles, until thoroughly cooked and deeply browned. Check that the meat mixture reaches at least 160°F with a food thermometer. Let any released moisture evaporate so the meat can brown. Spoon off excess grease, leaving about 1 tablespoon with the meat.
Add the yellow onion, bell pepper, and celery. Cook for 5–7 minutes, stirring frequently, until softened. Reduce heat to medium. Stir in the garlic, remaining 1½ teaspoons Cajun seasoning, smoked paprika, thyme, and black pepper. Cook for 30 seconds.
Add the beef broth, Worcestershire sauce, and optional hot sauce. Scrape up the browned bits and simmer uncovered for 3–5 minutes, until the meat mixture is moist and glossy with very little loose liquid. Stir in the butter.
Return the green beans to the wok and toss for 1–2 minutes. Gently fold in the potatoes and heat through uncovered. If the wok is too full to mix comfortably, combine in two batches. Taste, adjust seasoning, and finish with the sliced green onions.