Combine bulgur and warm water in a medium bowl; let stand until the bulgur is tender and the liquid is absorbed, about 30 minutes.
Prepare a grill or preheat the broiler. Add beef, watercress, salt and pepper to the plumped bulgur and mix thoroughly but lightly. Shape the mixture into eight 3/8-inch-thick patties (see Tip). Sandwich cheese between the patties to form 4 stuffed burgers.
Grill or broil the patties on a lightly oiled rack (see Tip) until browned and cooked through, 4 to 5 minutes per side. (An instant-read thermometer inserted in the center should register 160° F.) Place the burgers on buns and garnish with watercress sprigs.
When forming burger patties, make them thinner at the center to prevent them from steaming as they cook.To oil a grill: Oil a folded paper towel, hold it with tongs and rub it over the rack. (Do not use cooking spray on a hot grill.) When grilling delicate foods like tofu and fish, it is helpful to spray the food with cooking spray. Ingredient note: Bulgur is a quick-cooking form of wheat that has been parboiled and dried. Don't confuse it with cracked wheat, which needs longer cooking.