Place the water in a medium saucepan and bring to boiling over high heat. Add the bulgur; cover, reduce heat, and simmer for 10 to 12 minutes or until tender. Remove from heat. Let stand, covered, for 5 minutes. Uncover; fluff with a fork. Cool.
In a large mixing bowl combine the beef, onion, salt, pepper, and bulgur. Shape mixture into 6 1/2-inch thick patties.
For a charcoal grill, place the patties on the greased rack of an uncovered grill directly over medium coals. Grill for 10 to 12 minutes or until done (160°F), turning carefully once. (For a gas grill, preheat the grill. Reduce heat to medium. Place patties on the greased grill rack; cover and grill as above.)
Serve burgers in buns topped with lettuce, tomato, and avocado.