At home (up to two days in advance), thinly slice the red onion and place it in a resealable container. Add salt and the juice of two limes, shake, and refrigerate.
At camp, heat the cooking oil in a cast-iron skillet. Pat the chicken thighs dry and salt them on both sides. Add the thighs to the heated skillet and pan-fry until brown on one side, 6-8 minutes. Flip and continue cooking on the other side until done, 2-4 minutes, and the middle registers 165F.
Transfer the chicken to a cutting board and chop it into bite-sized pieces. Transfer the pieces to a bowl and add the barbecue sauce and cilantro. Drain the red onions and add half of them to the chicken mixture, reserving the rest. Toss to coat.
Wipe out your skillet and add the butter. For each quesadilla, assemble directly in the skillet: Bottom tortilla, ¼ cup of shredded cheese, half the chicken mixture, another ¼ cup of shredded cheese, and a second tortilla. Cook until the bottom is golden brown, 2-4 minutes, then flip. Cook 2-3 minutes more, until the second side is golden brown and the cheese is melted.
Top with more pickled red onions and cilantro. Serve with salsa and
