Heat gas or charcoal grill. In medium bowl, mix all ingredients except onion, bread and oil. Shape mixture into 4 patties, about 3/4 inch thick.
Place patties on grill over medium heat. Cover grill; cook 15 to 18 minutes, turning once, until no longer pink in center. For last 8 to 10 minutes of cooking, add onion slices; cook, turning once, until golden brown. Brush both sides of bread slices with oil. Add bread to grill for last 2 to 3 minutes of cooking; cook, turning once, until golden brown.
Serve burgers and onion between bread slices.