In a medium bowl, mix the shredded chicken with barbecue sauce until fully coated.
Heat 1 tablespoon olive oil in a large skillet over medium heat.
Place one tortilla in the skillet. Sprinkle ½ cup cheese over the tortilla, then spread ½ cup of the BBQ chicken mixture evenly. Top with another ½ cup cheese.
Place a second tortilla on top and cook for 3–4 minutes until the bottom is golden and the cheese starts to melt.
Carefully flip the quesadilla and cook another 3–4 minutes until the other side is golden and the cheese is fully melted.
Remove from the skillet and repeat with the remaining tortillas and filling.
Slice into wedges and garnish with chopped green onions if desired. Serve warm with sour cream or guacamole.