Preheat the oven to 375°F (190°C)
In a large skillet over medium-high heat, melt 1 tablespoon of butter. Brown the ground beef, breaking it into small pieces, until fully cooked. Drain excess fat and set aside.
In the same skillet, melt the remaining 1 tablespoon of butter. Add the diced onion and sauté for 3-4 minutes until softened. Add the minced mushrooms and cook for another 5 minutes or until the liquid has evaporated.
Combine the cooked beef, sautéed onion and mushrooms, ketchup, Worcestershire sauce, salt, and pepper in the skillet. Stir well to mix and cook for an additional 2-3 minutes.
Place the bell pepper halves in a baking dish, cut side up. Divide the beef and mushroom mixture among the peppers. Top each with 1/3 cup of shredded cheddar cheese.
Pour the beef broth into the baking dish to keep the peppers moist. Cover the dish with aluminum foil and bake for 35-40 minutes. Remove the foil and bake for an additional 20 minutes or until the peppers are tender and the cheese on top is golden and bubbly.
Remove from the oven, let sit for 5 minutes, then serve.