Insert the plates on their grill side. Preheat Griddler to high with the unit closed.
In a small bowl, combine the chicken, onion, and jalapeño peppers.
Sprinkle half the cheese in the center of each tortilla. Top with the chicken mixture, keeping the chicken mixture in a "log" about 1-1/2 inches wide and 4 to 5 inches long in the center of the tortilla.
Fold one side over the filling lengthwise to cover, fold top and bottom over short sides to cover, then fold last side over to close. Turn over so that flap is on bottom. Lightly brush the tops and bottoms of the wraps with the olive oil. Place quesadilla wraps on preheated Griddler. Close, using medium-light pressure. Grill/bake for 3-1/2 to 4 minutes, until tortilla is warm with grill markings, filling is warm, and cheese is melted.
Serve with salsa if desired.