1Place chicken in large heavy-duty resealable plastic bag. Stir together marinade ingredients; pour over chicken in bag. Seal and refrigerate at least 1 hour but no longer than 8 hours. Remove chicken from marinade; discard marinade. Heat gas or charcoal grill. Place chicken on grill over medium heat. Cover grill; cook 14 to 16 minutes, turning once, until juice of chicken is clear when center of thickest part is cut (at least 165°F). Thinly slice chicken.
2Meanwhile, stir together sauce ingredients. Cover and refrigerate until serving.
3To make quesadillas, top half of each tortilla with chicken, kimchi, cheese and green onion. Fold tortillas in half. Brush both sides of tortilla with oil. Heat 12-inch nonstick skillet over medium-low heat. Add 2 quesadillas to skillet. Cook 2 to 3 minutes, turning once, until tortilla is brown and crisp and cheese is melted. Repeat with remaining quesadillas. Cut quesadillas into wedges. Serve with sauce.