In a large skillet over medium-high heat, brown the ground beef, breaking it up as it cooks. Drain off any excess grease thoroughly.

Add the chopped onion and green bell pepper to the skillet with the browned beef. Cook until the vegetables begin to soften, about 5-7 minutes.

Transfer the cooked meat and vegetable mixture to the basin of a 4 or 6-quart slow cooker.

Stir in the garlic powder, Worcestershire sauce, dry mustard, salt, pepper, tomato sauce, ketchup, and brown sugar until all ingredients are evenly combined with the meat.

Cover the slow cooker and cook on LOW for 6 to 8 hours, or on HIGH for 3 to 4 hours, allowing the flavors to meld.

Before serving, stir the mixture well. For best flavor, let it simmer for 30 minutes after stirring. If desired, place cheese slices on buns and briefly broil or microwave to melt, then top with the Sloppy Joe mixture.