In a medium bowl, combine the rinsed black beans, corn, minced jalapeño, diced red onion, and minced garlic. In a small bowl, whisk together the fresh cilantro, apple cider vinegar, olive oil, honey, and lime juice to create a bright vinaigrette. Pour the vinaigrette over the bean mixture and stir until everything is evenly coated. Let this sit at room temperature while you prepare the chicken—the flavors will meld beautifully. I like to make this first so the acid from the vinegar can start softening the beans and marrying the flavors together.
In a medium bowl, combine the shredded or cubed chicken with the BBQ sauce and smoked paprika, stirring until the chicken is evenly coated and well incorporated. If the chicken is cold, you can heat this mixture gently in a skillet over medium heat for 2-3 minutes, stirring occasionally, just until warmed through. This step ensures the chicken is flavorful and at a good temperature to melt the cheese when assembled in the quesadillas.
Heat a large skillet over medium heat. Place one flour tortilla in the skillet and let it warm for about 30 seconds, then sprinkle roughly 1/3 cup of freshly shredded Monterey Jack cheese on the tortilla. Once the cheese begins to soften slightly (about 1 minute), layer about 1/2 cup of the seasoned chicken from Step 2 and 1/2 cup of the cowboy caviar from Step 1 on top of the cheese. Sprinkle another 1/3 cup of cheese over the filling, then top with a second tortilla. I find that loading the cheese on both sides helps create a beautiful, melted exterior that holds everything together.
Cook the assembled quesadilla for 2-3 minutes without moving it, allowing the bottom tortilla to turn golden brown and the cheese to start melting. Carefully flip the quesadilla using a wide spatula and cook the other side for another 2-3 minutes until it's also golden brown and all the cheese is fully melted. The cheese should be slightly oozing out the sides—this is exactly what you want. Repeat this process with the remaining 3 tortillas and filling to make 4 quesadillas total.
Transfer each cooked quesadilla to a cutting board and cut it into 4 triangular pieces using a sharp knife or pizza cutter. Serve immediately while the cheese is still warm and melted. The quesadillas are best enjoyed right off the skillet when they're at their crispiest and most delicious.