To prepare sauce, combine first 3 ingredients, stirring with a whisk; set aside.
To prepare quesadillas, combine peaches, 1 tablespoon chives, and sugar, tossing gently to coat. Heat a large nonstick skillet over medium-high heat. Arrange one-fourth of cheese and one-fourth of peach mixture over half of each tortilla; fold tortillas in half. Coat pan with cooking spray. Place 2 quesadillas in pan; cook 2 minutes on each side or until tortillas are lightly browned and crisp. Remove from pan; keep warm. Repeat procedure with remaining quesadillas. Cut each quesadilla into 3 wedges; serve with sauce. Garnish with chive strips, if desired.
Placing the fillings on one side of the tortilla and folding the other half over (like a taco) makes the quesadillas easier to handle. Whole wheat tortillas would also be great, offering a sweet, nutty taste. The lime-honey dipping sauce balances the richness of the buttery filling.