Follow these detailed steps carefully to ensure your quesadillas are perfectly cooked, with a crispy outside and a hot, melted inside.

1. Prepare the Filling: In a large mixing bowl, combine the shredded chicken and your chosen BBQ sauce. Stir well until every piece of chicken is evenly coated in the sauce. This step is crucial for ensuring every bite is packed with that smoky BBQ flavor.
2. Incorporate the Vegetables and Herbs: Add the corn kernels (fresh or frozen), diced red onion, and chopped fresh cilantro to the bowl with the BBQ chicken. Mix everything together until all the ingredients are well-distributed. This mixture is the flavor core of your quesadillas.
3. Combine the Cheeses: In a separate, smaller bowl, toss the shredded Monterey Jack and shredded sharp cheddar cheese together. Mixing them beforehand ensures you get a perfect blend of both cheeses in every bite, leading to a more complex flavor and ideal melt.
4. Assemble the Quesadillas: Lay a large flour tortilla flat on a clean work surface. Sprinkle about ¼ cup of the cheese mixture evenly over one half of the tortilla, leaving a small border around the edge. Spoon approximately ½ cup of the BBQ chicken and corn mixture over the cheese. Top the chicken mixture with another ¼ cup of the cheese blend. This cheese-filling-cheese layering technique is the secret to a well-structured quesadilla that holds together.
5. Fold and Prepare for Cooking: Carefully fold the empty half of the tortilla over the filling to create a half-moon shape. Gently press down to seal it and evenly distribute the filling. Repeat this assembly process for the remaining tortillas and filling. You should have 8 assembled quesadillas ready for the skillet.
6. Cook the First Side: Heat a large non-stick skillet or a cast-iron pan over medium heat. Once the pan is hot, add about ½ tablespoon of butter and let it melt and coat the surface. Carefully place one or two assembled quesadillas in the pan, ensuring not to overcrowd it. Cook for 2-4 minutes on the first side, or until the bottom is a deep golden-brown and crispy. You can check by gently lifting an edge with a spatula.
7. Flip and Finish Cooking: Using a wide spatula, carefully flip the quesadilla. Cook for another 2-3 minutes on the second side, until it is equally golden-brown and the cheese inside is completely melted and oozing.
8. Rest and Serve: Transfer the cooked quesadilla to a cutting board and let it rest for a minute. This allows the cheese to set slightly, which prevents the filling from spilling out when you cut it. Repeat the cooking process with the remaining butter and quesadillas. Cut each quesadilla into 2 or 3 wedges and serve immediately while hot.