Preheat a gas or charcoal grill or heat a large skillet over medium-high heat.
Combine the beef, horseradish, garlic, salt, and pepper in a medium mixing bowl. Mix gently but thoroughly. Divide the mixture into 4 equal portions, form into balls, and flatten slightly to form 3/4-inch-thick patties.
Grill the burgers or cook in the skillet for 2 minutes. Flip the patties over and place 1 slice of cheese on each patty. Cook for 2 minutes more for medium-rare burgers, or longer if you wish. Transfer to a platter.
Spread both cut sides of each roll with the butter. Toast them on the grill or in the skillet for about 30 seconds. Put the burgers on the buns and serve immediately, with the condiments.
Yield: 4 servings