Add ground beef to a nonstick pan placed over medium heat. Use a wooden spoon to break it apart as it cooks. As soon as you have it crumbled, add the onion and green pepper. Cook until beef is no longer pink and onion and pepper is softened.
Add ketchup, dry mustard, garlic powder, Worcestershire sauce, and cayenne pepper. Stir to combine well and cook over medium-low heat for 5 minutes. Remove from heat.
For each sandwich, spread 1/4 cup cheese on top of one slice of bread. Top with sloppy joe mixture. Place second piece of bread on top.
Heat griddle or nonstick pan over medium heat. Add 1/2 tablespoon butter, let it melt and place sandwich on top. Cook until bottom is golden, lift sandwich with a spatula, place another 1/2 tablespoon of butter down, and flip sandwich over. Cook until cheese is melted.