In a large skillet over medium heat, cook the ground beef until browned, breaking it apart with a spatula, about 5-7 minutes.
Add the chopped onion and minced garlic to the skillet and cook until the onion is translucent, about 3 minutes.
Stir in the ketchup, tomato paste, Worcestershire sauce, smoked paprika, chili powder, salt, and black pepper. Simmer the mixture for 8-10 minutes, stirring occasionally, until thickened.
While the beef mixture simmers, preheat the oven broiler.
In a small bowl, combine the softened butter and minced garlic to make garlic butter.
Spread the garlic butter evenly on one side of each bread slice.
Place the bread slices buttered side up on a baking sheet and broil for 1-2 minutes until golden and toasted. Watch carefully to avoid burning.
Flip the toasted bread slices over and divide the sloppy joe beef mixture evenly on the untoasted sides.
Sprinkle shredded cheddar cheese evenly over the beef on each slice.
Return the baking sheet to the oven and broil for another 2-3 minutes until the cheese is melted and bubbly.
Remove from the oven and serve immediately.