Cut each chicken breast in half and slightly flatten with palm of hand. Peel melons and slice into thin strips (or cut into melon balls). Place vegetable oil in large frypan over medium heat. Sprinkle chicken with salt and pepper and add to pan; sauté about 5 minutes on each side until golden brown. Add sake (or wine), chicken broth and ginger; simmer for several minutes. Dissolve cornstarch in rice wine vinegar. Remove chicken to serving platter. To frypan, add vinegar mixture, honey and melons, stirring gently. Add butter and parsley; dribble sauce over chicken.