In a large bowl, combine the ground turkey, feta, olives, red onion, bell pepper, parsley, dill, 1 tablespoon olive oil, garlic, salt, paprika, cumin, oregano, and black pepper. Mix with a fork or use your hands to combine, being careful not to overmix.
Divide the turkey mixture into 4 equal portions, then shape each one into a ½-inch-thick patty. Refrigerate, uncovered, for at least 20 minutes and up to 4 hours.
Heat the remaining 1 tablespoon olive oil in a grill pan or cast iron skillet over medium heat. Cook the patties for 5 to 6 minutes per side, until they reach an internal temperature of 165°F.
Alternatively, grill the burgers over direct medium heat for 4 to 6 minutes per side, until cooked through and the centers reach 165°F.
Serve the burgers on buns, in pita, or in lettuce wraps. Top with tzatziki or tahini and your favorite vegetables.