In a wide pan over medium heat, add ground beef. Cook, breaking apart with the back of the spoon until it begins to render fat.
Drain any excess fat except for about 1 tablespoon. Add onions and bell peppers and continue to cook until softened and meat is lightly browned.
Add tomato sauce, ketchup, Worcestershire sauce, yellow mustard, vinegar, and brown sugar. Stir until wel incorporated and sugar is dissolved. Bring to a boil.
Lower heat, cover, and simmer, occasionally stirring, for about 15 to 20 minutes or until meat is cooked and sauce is thick and reduced.
Season with salt and pepper to taste. Remove from heat and allow to slightly cool.
Preheat oven to 350 F.
Spray a 9x13 baking pan with cooking spray.
Using a serrated knife, cut the rolls in half horizontally and place the bottom slab of bread in the prepared baking dish.
Spoon sloppy joe mixture on the bread and spread evenly to cover the surface.
Arrange cheese slices over the beef mixture.
Place the top slab of the bread over the cheese.
In a small bowl, combine melted butter, Parmesan cheese, and Italian seasoning. Brush over the rolls.
Cover with foil and bake in the oven for about 10 to 15 minutes or until hot and cheese is melted.
Remove from heat and cut into individual pieces to serve.