Place a nonstick skillet over medium heat and add the ground beef. Break it apart with a wooden spoon during cooking. Once it begins to crumble, incorporate the chopped onion and green pepper. Continue cooking until the beef is no longer pink and the vegetables have softened.

Incorporate ketchup, Worcestershire sauce, ground mustard, garlic powder, and cayenne pepper into the pan. Stir until well blended and allow the mixture to simmer over medium-low heat for 5 minutes. Remove from heat.

To assemble, layer approximately 1/4 cup shredded cheddar over one slice of bread. Spoon a generous portion of the sloppy joe mixture over the cheese, then cap with another slice of bread.

Preheat a griddle or nonstick skillet over medium heat. Melt 1/2 tablespoon of butter, then set the sandwich in the pan. Grill until the underside is golden brown, lift with a spatula, add another 1/2 tablespoon butter, and flip. Continue to cook until the second side is equally golden and the cheese is thoroughly melted.