Shred the rotisserie chicken into bite-size pieces, discarding any skin and bones. In a large bowl, toss the chicken with the 1/2 cup BBQ sauce until every piece is evenly coated. Thinly slice the red onion and chop the cilantro.

Set the large skillet over medium-high heat. Add 1/2 tablespoon of the vegetable oil and swirl to coat; let it shimmer. While it heats, lay out two of the tortillas on the counter.

Spread one-quarter of the BBQ chicken over one half of each tortilla. Top each with 1/4 of the cheese blend, a few red onion slices, and a pinch of cilantro. Fold the empty half over the filling and press gently so the tortillas hold their shape.

Lay the two folded quesadillas in the hot skillet. Cook 3 minutes per side, pressing lightly with the spatula, until the tortillas are deep golden and crisp and the cheese is fully melted. Transfer to a cutting board.

Repeat: add another 1/2 tablespoon oil to the skillet, fill and fold the remaining two tortillas, and cook 3 minutes per side until golden and crisp. If the skillet looks dry between batches, add the remaining oil.

Let the quesadillas rest 1 minute, then slice each into 4 wedges. Serve hot with ranch dressing or extra BBQ sauce for dipping.