In a medium skillet over medium heat, cook the ground beef and chopped onion until the beef is browned and the onion is soft, about 5-7 minutes. Drain any excess fat.
Add ketchup, tomato paste, Worcestershire sauce, yellow mustard, garlic powder, smoked paprika, salt, and black pepper to the skillet. Stir to combine and simmer for 5 minutes until the mixture thickens slightly. Remove from heat.
Heat a clean skillet or griddle over medium heat.
Butter one side of each bread slice evenly.
Place 4 slices of bread, buttered side down, on the skillet. Layer each with one slice of cheddar cheese, then evenly divide the sloppy joe mixture over the cheese, and top with another slice of cheddar cheese.
Top each sandwich 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 1-2 minutes, then slice and serve warm.