In a bowl, combine 2 cups cooked shredded chicken with 1/2 cup barbecue sauce. Mix well to coat. Finely chop 1/4 cup green onions (if using).
Heat 1/2 tablespoon olive oil in a large non-stick skillet over medium heat. Place one large flour tortilla in the pan. Sprinkle about 1/4 cup shredded cheddar cheese evenly over one half of the tortilla. Top with 1/4 of the BBQ chicken mixture and a portion of the green onions. Sprinkle with another 1/4 cup shredded cheddar cheese. Fold the empty half of the tortilla over the filled half to create a half-moon.
Cook the quesadilla for 3-5 minutes per side, or until the tortilla is golden brown and crispy, and the cheese is completely melted and gooey. Remove from the skillet and place on a cutting board.
Wipe out the skillet if needed, add a little more oil, and repeat the assembly and cooking process for the remaining tortillas and filling. Let each quesadilla rest for a minute, then slice into 2-3 wedges. Serve immediately with desired toppings like sour cream, salsa, or extra BBQ sauce.