In a small bowl stir together mayonnaise and basil. Pit and chop olives. In a large bowl blend together olives, turkey, tomatoes, thyme, and salt and pepper, to taste. Form mixture into 4 (1-inch-thick) patties. When forming patties, handle meat as little as possible for a juicier burger. Prepare grill.
Grill burgers on an oiled rack set 5 to 6 inches over glowing coals about 4 1/2 minutes on each side for well-done. (Alternatively, burgers may be cooked in a hot well-seasoned cast-iron skillet or ridged grill pan.) Make sure burgers are cooked through. Toast sesame buns. Transfer burgers to buns and top with basil mayonnaise and arugula.
Recipe courtesy of Gourmet Magazine