Heat gas or charcoal grill. In medium bowl, mix beef, 1/4 cup of the dressing, the egg and bread crumbs. Shape into 5 oval patties, about 1/2 inch thick.
Place patties on grill over medium heat. Cover grill; cook 7 to 10 minutes, turning once, until thermometer inserted in center of patties reads 160°F. During last 1 to 2 minutes of cooking time, top each patty with cheese.
If desired, place pita bread halves on grill during last 1 minute of cooking. Fill pita bread halves with remaining 1/4 cup dressing, tomato, cucumber and patties.