Prepare a charcoal grill and position the rack 4 inches above medium-high heat, or heat a gas grill to medium-high heat. Lightly oil the grill.
Form the beef into 6 patties, about 3 1/2 inches in diameter, and sprinkle on both sides with the salt and pepper. Grill the burgers, charring well on both sides, until almost but not quite as done as desired, 3 to 4 minutes per side for medium-rare. (Gas grills may require 1 to 2 minutes more total cooking.) Transfer the burgers to a rack to rest for 5 minutes. Top each burger with about 1/2 tablespoon of the butter and a sprinkling of the chopped parsley. Meanwhile, toast the English Muffins on the grill for 2 to 3 minutes.
Reheat the burgers on the grill for about 1 minute. Serve them on the toasted English muffins, topped with the tomato slices.