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 apart with a spatula, until browned and cooked through, about 6-7 minutes.
Drain excess fat from the skillet if necessary.
Stir in ketchup, tomato paste, Worcestershire sauce, yellow mustard, garlic powder, smoked paprika, salt, and black pepper.
Simmer the sloppy joe mixture for 5 minutes, stirring occasionally, until thickened. Remove from heat.
Preheat a separate large skillet or griddle over medium heat.
Butter one side of each bread slice with softened butter.
Place four bread slices, buttered side down, on the skillet.
Layer each slice with one slice of cheddar cheese, then evenly distribute the sloppy joe mixture over the cheese.
Top each with another slice of cheddar cheese, then place the remaining bread slices on top, 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 sandwiches from skillet, let cool for 1 minute, then slice in half and serve immediately.