Combine first 7 ingredients. Add chicken; mix lightly but thoroughly. Gently stir in cheese., Shape into four 1/2-in.-thick patties (mixture will be soft). Brush patties with oil. Grill, covered, over medium heat until a thermometer reads 165°, 5-6 minutes per side. Serve on buns. If desired, top with tzatziki sauce and sliced tomato.