Heat vegetable oil in a large skillet over moderate heat. Add diced onions and cook until they begin to brown, approximately 6 minutes.
Push the onions to one side of the skillet and add ground sirloin and chuck. Cook, stirring occasionally, until browned, about 12 minutes.
Add minced garlic to the skillet and cook until fragrant, about 1 minute.
Stir in diced tomatoes, ketchup, barbecue sauce, tomato paste, and cider vinegar. Simmer the mixture over moderate heat, stirring occasionally, until the sauce thickens, approximately 20 minutes.
Season with salt and pepper to taste.
Spoon the barbecue sloppy joe mixture onto toasted hamburger buns. Optionally, top with sliced sharp cheddar cheese.
Serve and enjoy your homemade barbecue sloppy joes!