Heat olive oil in a large skillet over medium heat.
Add diced 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 apart with a spoon, until browned and cooked through, about 6-7 minutes.
Drain excess fat from the skillet.
Stir in ketchup, tomato paste, Worcestershire sauce, yellow mustard, garlic powder, onion powder, smoked paprika, 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 slice of bread.
Place four slices of bread, buttered side down, on the skillet.
Top each with a slice of cheddar cheese, then evenly divide the sloppy joe mixture over the cheese.
Place another slice of cheddar cheese on top of the sloppy joe mixture, then cover with the remaining bread slices, buttered side up.
Cook sandwiches for 3-4 minutes on each side, pressing gently with a spatula, until the bread is golden brown and the cheese is melted.
Remove from skillet, let cool for 1 minute, then slice in half and serve.