Preheat your BBQ grill to medium-high heat.
Place the chicken breasts between two sheets of plastic wrap. Using a meat mallet or rolling pin, gently pound the chicken breasts to flatten them to an even thickness. This will help them cook more evenly.
Drizzle the flattened chicken breasts with olive oil and rub it in to coat both sides.
Sprinkle the Revolution Barbecue All Purpose Rub over both sides of the chicken breasts, pressing the rub into the meat to ensure it adheres well.
Place the seasoned chicken breasts on the preheated grill and cook for about 6-8 minutes per side, or until the internal temperature reaches 165°F (74°C). Cooking time may vary depending on the thickness of the chicken breasts. Once cooked, remove from the grill and let them rest for a few minutes before slicing.
While the chicken is resting, prepare the avocado crema. In a blender or food processor, combine the ripe avocado, sour cream, lime juice, and a pinch of salt and pepper. Blend until smooth and creamy. Taste and adjust the seasoning if needed.
Heat a large skillet or griddle over medium heat. Place a flour tortilla on the skillet and sprinkle a layer of shredded Mexican quesadilla cheese on one half of the tortilla.
Slice the grilled chicken breasts into thin strips and place a generous amount on top of the cheese.
Fold the tortilla in half to cover the filling and press it down gently with a spatula.
Cook the quesadilla for about 2-3 minutes per side, or until the tortilla is golden brown and the cheese has melted.
Repeat the process with the remaining tortillas, cheese, and chicken.
Once all the quesadillas are cooked, cut them into wedges or triangles.
Serve the BBQ Chicken Quesadillas with the avocado crema