Heat large skillet over medium-high heat and add olive oil. Add peaches and cook for 2-3 minutes or until slightly caramelized. Remove and set aside.
Add another tablespoon of olive oil to skillet. Salt and pepper chicken and add to skillet and cook for about 6-8 minutes, stirring occasionally until cooked through. Shred chicken with fork and add 3-4 tablespoon BBQ sauce until coated. Stir in the jalapeño, green pepper and cilantro.
Place tortilla on plate and add ½ cup shredded monterey jack cheese. (or more if you desire) Top with chicken mixture and peaches. Try to use about a ½ of each as this makes 2 quesadillas. Fold the tortilla and repeat.
Heat butter in large skillet over medium heat. Add both quesadillas to skillet and cook on one side for about 5 minutes or until browned. Flip both quesadillas and cook for another 5 minutes or until browned. Use the pizza cutter to cut into four slices.
Flour Tortillas: While corn tortillas are great, you need flour tortillas for the pliability and ability to hold the cheese and filling together.
Use One Tortilla: Making a quesadilla with two tortillas can be difficult to flip. Using the half-moon technique makes it much easier to flip. Just spread the filling over half the tortilla, then fold the other half over the filling.
Don’t Overstuff It: A quesadilla is best when you keep it on the lighter side. You want enough filling to hold everything together.
Freshly Grated Cheese: don’t use pre-shredded cheese. It doesn’t taste as good and also has additional ingredients that you don’t need.
Go Easy on the Butter/Oil: Flour tortillas absorb butter and oil easily, so you want to just lightly coat the pan for these chicken quesadillas.
Cook on Medium Heat: Tortillas burn easily, so don’t cook on high heat. You’ll want to cook the quesadilla on medium-low to medium heat.