In a medium skillet combine chicken and the water. Bring to boiling; reduce heat. Simmer, covered, for 10 to 12 minutes or until chicken is no longer pink (170 degrees F), turning once; drain. Using two forks, pull chicken apart into shreds (you should have about 1 1/3 cups). In a small bowl toss together shredded chicken and hot pepper sauce to combine.
Coat one side of each tortilla with cooking spray. Place tortillas, coated sides down, on a work surface. Top half of each tortilla with mozzarella cheese, blue cheese, and shredded chicken. Fold tortillas in half, pressing gently.
In a large skillet cook quesadillas, half at a time, over medium heat for 4 to 5 minutes or until golden and cheeses are melted, turning once. Cut into wedges. Serve with carrots, celery, and salad dressing.