Warm a large skillet over medium heat and add the olive oil.
Soften the onion and bell pepper for 4 to 5 minutes, stirring now and then.
Stir in the garlic and cook until fragrant, about 30 seconds.
Add the ground beef. Cook, breaking it up, until browned. Drain excess fat if needed.
Stir in tomato sauce, ketchup, tomato paste, brown sugar, Worcestershire, mustard, vinegar, chili powder, paprika, salt, pepper, and broth.
Simmer uncovered for 8 to 10 minutes, stirring, until thick and glossy.
Fold in 1 cup of the cheddar until melted. Keep the rest for topping.
Butter the buns and toast in a skillet or under the broiler until golden.
Pile the meat onto buns, sprinkle with remaining cheese, cover a minute to melt, and serve.