Set one flour tortilla on a preheated cast iron griddle. Sprinkle an even layer of shredded Mexican-blend cheese across the surface.
Distribute the shredded chicken, followed by the diced jalapeño and red onion, over the cheese. Lay the second tortilla on top to form a sandwich.
Cook over medium-low heat until the underside becomes crisp and golden, approximately 7 to 10 minutes. Flip carefully, then continue cooking the opposite side until equally golden, around 3 to 5 minutes more.
Once both sides are crisped, remove the quesadilla from the griddle and place it on a cutting board. Slice into triangles and serve hot.