To grill pineapple: cut wedges, and grill over low heat. Cut into slices.

To pound chicken, place inside a Ziploc bag and pound with a mallet or large can to flatten to uniform thickness.

Sprinkle with salt and pepper and Cajun spice (or cumin/cayenne, etc.) Grill on outdoor grill, brushing with barbecue sauce. Set aside and slice into very thin slices.
Warm griddle over medium heat and put sizzling butter in the pan. Slightly warm and toast tortillas on both sides, then remove from griddle and set aside.
To assemble, sprinkle four browned tortillas with grated Monterey Jack. Arrange chicken slices evenly over the surface. Add pineapple slices. Sprinkle on cilantro. If desired, drizzle extra barbecue sauce over the top. Add second browned tortilla on top of each one.
Add to oven to warm, or back to skillet to warm, until cheese is melted. Cut each quesadilla into six wedges. Serve with sour cream, pico de gallo, and lime wedges.