In 2- quart saucepan, heat chicken and Buffalo wing sauce to boiling over medium-high heat. Reduce heat to low; cover. Simmer 5 to 7 minutes or until chicken is cooked through.
Meanwhile, in 10-inch nonstick skillet, melt butter over medium heat. Add onion; cook, stirring frequently, 4 to 6 minutes or until starting to brown. Stir in sugar; reduce heat to low and cook, stirring frequently, 6 minutes longer or until onion is caramelized. Set aside.
Heat closed contact grill 5 minutes.
Meanwhile, microwave each pita covered with damp paper towel on High 20 to 25 seconds or until soft enough to fold without breaking. Lay pitas flat on work surface. Place one-fourth of the caramelized onion on half of each pita. Using slotted spoon, spoon about 1/3 cup chicken over onion. (Reserve liquid in saucepan to serve with Panini). Sprinkle with 1 tablespoon of the cheese. Carefully fold pita over filling. Repeat with remaining pitas.
When grill is heated, place sandwich on grill. Close grill; grill 4 to 5 minutes or until cheese is melted and pita is toasted. Serve panini with reserved sauce.