Heat olive oil in a large skillet over medium heat.
Add diced onion and green bell pepper to the skillet and cook for 3-4 minutes until softened.
Add ground beef to the skillet and cook until browned, breaking it apart with a spatula, about 5-6 minutes.
Drain excess fat from the skillet.
Stir in ketchup, tomato paste, Worcestershire sauce, yellow mustard, garlic powder, onion powder, salt, and black pepper.
Simmer the mixture for 5 minutes, stirring occasionally, until thickened. Remove from heat.
Preheat a clean skillet or griddle over medium heat.
Butter one side of each bread slice.
Place 4 slices of bread, buttered side down, on the skillet.
Layer one slice of cheddar cheese on each bread slice in the skillet.
Evenly distribute the Sloppy Joe mixture over the cheese.
Top each sandwich with another slice of cheddar cheese and a second slice of bread, buttered side up.
Cook for 3-4 minutes until the bottom bread is golden brown and the cheese begins to melt.
Carefully flip each sandwich and cook for another 3-4 minutes until the other side is golden brown and the cheese is fully melted.
Remove sandwiches from the skillet, let cool for 1-2 minutes, then slice and serve.