Meanwhile, in 1-quart saucepan, heat 1/3 cup vinegar to boiling over medium heat. Add brown sugar; cook 3 to 5 minutes, stirring constantly, until vinegar mixture is reduced and slightly thickened. Remove from heat; set aside. Cut pocket in each chicken breast by making horizontal slit along thin long edge, being careful to not cut through to opposite side. Stuff each pocket with 1 slice mozzarella and 1 tablespoon pesto. Press edges of chicken together to seal. Season chicken with 3/4 teaspoon of the salt and remaining 1/2 teaspoon pepper. Reserve 1 tablespoon of the vinegar glaze to brush on at the end.