Cut each chicken breast in half; place into large resealable plastic food bag. Flatten each to about 1/2-inch thickness with meat mallet.
Pour desired amount of buffalo wing sauce into bag; seal bag. Gently shake to coat chicken; refrigerate at least 30 minutes.
Heat gas grill on medium or charcoal grill until coals are ash white.
Remove chicken from hot sauce; discard sauce.
Place chicken onto grill. Grill, turning once, 10-12 minutes or until internal temperature reaches at least 165 degrees F and chicken is no longer pink in center. Brush with additional buffalo wing sauce while cooking, if desired. Top each chicken piece with 2 quarters cheese during last minute of cooking.
Place buns, cut-side down, onto grill. Cook 1-2 minutes or until toasted.
Spread dressing onto cut-sides of toasted buns. Place celery sticks and chicken onto bottom half of buns; cover with top half of buns.