Shred chicken and mix with BBQ sauce, shred cheese, chop onion, and butter one side of each tortilla.
Cook onions in olive oil to soften, then mix with shredded chicken and BBQ sauce. Set aside.
Heat a large skillet and place one tortilla, butter side down, in the skillet.
Layer cheese on half of the tortilla, then add chicken filling, and top with more cheese. Fold tortilla in half over the filling and press down gently.
Cook until golden brown, then flip with a rubber spatula. Cook until the other side is golden brown and cheese is melted.
Remove the cooked quesadilla from the skillet and place on a cutting board. If making several, keep them warm on a cookie sheet in the oven at the lowest temperature.
Let the quesadilla cool for at least 5 minutes before slicing and serving with optional sides like cilantro, sour cream, guacamole, salsa, or ranch dip.