Brown and crumble the ground beef in a medium skillet until no longer pink. Drain if necessary.
Add the onions, green pepper, brown sugar, mustard, vinegar, salt, and ketchup, and simmer for 20-30 minutes or until the sauce reaches your desired consistency. The sauce will thicken more the longer it cooks.
Add the Velveeta to the beef mixture, stirring until the cheese is melted.
Remove the skillet from the heat and stir in half of the Cheddar until melted and well combined.
Serve the beef mixture on hamburger buns with the remaining Cheddar.
Notes
You can use ground chicken, turkey, or pork in place of ground beef.