In a large, high-sided skillet, add beef (1 pound), 3/4 teaspoon salt, and 1/4 teaspoon black pepper, then cook and crumble the meat until there's no longer any pink in the beef. Transfer to a plate. Drain and discard all but 2 tablespoons of fat,
Add 2 tablespoons of butter to the skillet then add potatoes (1 pound), onion (1 cup), red pepper, and carrot (1/4 cup). Stir often, flipping the vegetables every few minutes until the carrots and peppers become tender and the potatoes are almost cooked through, and soft when pierced with the tip of a knife, about 10-12 minutes. (If at 10 minutes the potatoes are not almost done, cover the skillet and continue to cook, until the potatoes are fork tender. You may need to lower the heat).
Sprinkle with Cajun seasoning (3/4 teaspoon) and smoked paprika (1/2 teaspoon) and toss until all the vegetables are evenly coated. Add garlic (1 heaping tablespoon) and continue to cook, stirring and flipping the mixture, for about 1 minute.
Return cooked beef to the skillet. Sprinkle 1/4 cup flour over skillet. Toss to evenly coat everything with flour.
Pour in the beef broth (1 1/2 cups) and Worcestershire sauce (1 tablespoon) over the meat and vegetable mixture. Cook over low, covered, stirring occasionally, until potatoes are tender, about 2-4 minutes. Add more beef broth if needed (careful not to thin the gravy too much!)
Uncover, sprinkle with cheese (1 /2cup), set heat on low, and cover to allow the cheese to melt, about 1-2 minutes. Sprinkle with sliced green onions (1/4 cup) and serve.