In a medium bowl beat egg white with a whisk; stir in bread crumbs, milk, and half of the dry salad dressing mix. Reserve remaining half of dry salad dressing mix for Olive-Tomato Relish. Add chicken; mix well. Shape chicken mixture into four 3/4-inch-thick patties.
For a charcoal grill, grill patties on the rack of an uncovered grill directly over medium coals for 14 to 18 minutes or until no longer pink (165 degrees F), turning once halfway through grilling.
Serve the burgers on pita bread rounds with Olive-Tomato Relish and feta cheese.
In a small bowl stir together vinegar, olive oil, and the remaining half of the dry salad dressing mix. Stir in tomato, cucumber, and olives.