Heat grill to medium-high. In a medium bowl, place beef, horseradish, Worcestershire sauce, 1 teaspoon coarse salt, and 1/4 teaspoon pepper. Mix very gently with a fork to combine, being careful not to overwork meat (which would make burgers dense).
Gently form meat into 4 equal-size patties, 4 inches in diameter and 1 inch thick. With thumb, make a 1/2-inch-deep indentation in the center of each patty (this helps keep burgers flat during grilling).
Moisten a folded paper towel with oil; grasp with tongs and rub over grates. Season burgers with salt and pepper. Grill 4 to 7 minutes per side for medium-rare, placing cheese on second side for last 3 to 4 minutes of cooking.
Serve on English muffins (toasted on grill, if you like), with desired toppings.