For burgers, gently combine chicken, 1 cup fresh cilantro or parsley plus dried cilantro, chopped onion, garlic, salt, and pepper in a large bowl. Shape into four 3/4-inch-thick patties. Brush tops of burgers with 3 teaspoons of the oil.
Heat remaining oil in a large grill pan or cast-iron skillet over medium-high heat. Place burgers, oiled sides down, on hot grill pan or skillet; cook for 8 minutes. Turn and cook for 8 to 10 minutes more or until internal temperature registers 165 degree F on an instant-read meat thermometer. Transfer burgers to a serving plate.
Meanwhile, combine mayonnaise, 1 cup fresh cilantro or parsley plus dried cilantro, orange peel, and juice in a small bowl. Spread mayonnaise mixture on cut sides of rolls. Top bottoms of rolls with lettuce, burgers, onion slices (if desired), and tops of rolls. Makes 4 servings.