While preparing, heat grill. In a blender, combine diced avocados, mayo, lime juice, ¼ cup of parsley, cayenne pepper and salt to taste. Blend until smooth. Set aside.
On a medium grill, lay out 6 flour tortillas (you can work with one at a time if you like or if space is limited on your grill). Top one half of the tortilla evenly with ingredients and in this order: mozzarella cheese, bacon, blue cheese, tomato slices, avocado slices and parsley. Fold empty side of tortilla over the topping side. Grill just a couple of minutes until cheese is melted. This will cook fast. Carefully flip each quesadilla with a spatula and tongs. Cook an additional 2 minutes. Remove from grill. Allow to set for 2 minutes. Cut each quesadilla into 3 slices and serve with Avocado Mayo for dipping.