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.