1. Chop half of the cucumber into small pieces, and place in a small bowl. Slice remaining cucumber into thin slices for sandwiches, and set aside. Add the mint, dill, lemon juice, garlic, and Greek yogurt to a small bowl, and refrigerate until ready to use.
2. In a large bowl, combine the feta, egg, spinach, olives, cornstarch, oregano, garlic, salt and pepper. Add the chicken, and mix lightly but thoroughly. Shape into equally-sized burger patties.
3. Moisten a paper towel with oil, and, using cooking tongs, rub on grill rack to coat lightly. Add burgers, and grill covered, over medium heat, until meat thermometer reads 165°F (about 5-7 minutes on each side).
4. Serve on iceberg lettuce “buns” with tomato slices, remaining cucumber, and sauce.