How to Make It
In a large skillet, heat the oil. Add the onion and cook over moderately high heat until beginning to brown, about 6 minutes. Push the onion to one side of the skillet and add the ground sirloin and chuck. Cook, stirring, until browned, about 12 minutes. Add the garlic and cook until fragrant, 1 minute. Stir in the tomatoes, ketchup, barbecue sauce, tomato paste and vinegar. Simmer over moderately low heat, stirring, until the sauce is thick, 20 minutes. Season with salt and pepper.
Preheat the broiler. Set the bun bottoms on serving plates. Lay the cheese on the bun tops and broil until melted, about 1 minute. Spoon the Sloppy Joe mixture on the buns; serve.