Heat olive oil in a large skillet over medium heat.
Add chopped onion and green bell pepper to the skillet and sauté for 3-4 minutes until softened.
Add ground beef to the skillet and cook, breaking it up with a spoon, until browned and cooked through, about 6-7 minutes.
Drain any 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 1 slice of cheddar cheese on each bread slice in the skillet.
Divide the Sloppy Joe mixture evenly over the cheese slices.
Top each with another slice of cheddar cheese.
Place the remaining bread slices on top, buttered side up.
Cook sandwiches for 3-4 minutes per side, pressing gently with a spatula, until the bread is golden brown and the cheese is melted.
Remove sandwiches from the skillet, let cool for 1 minute, then slice in half and serve.