Divide the beef into 4 equal portions. Press a cube of pepper jack into the center of each and shape the meat around the cheese; form into patties, about 4 inches wide and 3/4 inch thick. Season the patties with salt and pepper.
Heat the vegetable oil in a large skillet over medium-high heat. Add the patties and cook until browned on the bottom, about 4 minutes. Flip the patties and top each with a slice of Cheddar; cook 4 to 5 more minutes.
Meanwhile, toast the hamburger buns and spread with ketchup, mustard and/or mayonnaise. Fill with the burgers and top with lettuce, tomato and/or red onion. Serve with pickles.