In a bowl, combine the Cheddar and onions. Set aside until ready to use.
Mix together the ground beef, pepper, salt, garlic, oregano, cumin and egg until thoroughly combined, but not overworked. Form into 4 equal-sized balls. Flatten out each ball to 1/4-inch thick and put a tablespoon each of the onion and Cheddar in the middle. Fold the burger meat over the onion and Cheddar, enclosing it securely in the middle of the meat and form it back into a ball. Flatten into 3/4-inch thick burger patties. Grill for 5 to 6 minutes turning only once.
Toast the buns on the grill over indirect heat. Add the burgers to the buns and serve with the ketchup.