Heat grill or grill pan to medium-high. Form the beef into four ½-inch-thick patties and season with ½ teaspoon salt and ¼ teaspoon pepper.
Grill the burgers to the desired doneness, 4 to 5 minutes per side for medium. Baste with the teriyaki sauce during the last 2 minutes of cooking.
Grill the English muffins until toasted, 1 to 2 minutes per side.
Sandwich the burgers in the muffins. Serve with the tomatoes and cucumbers.