Cut 16 inch length of heavy duty foil;
Arrange mushrooms and onion on one half.
Drizzle with 1 tbsp of balsamic vinegar and butter;
Sprinkle with sage and half of salt and pepper.
Fold foil over vegetables; fold in sides and seal to form packet.
Place on grill over medium heat; close lid and cook, turning once, until tender, 10 minutes.
Meanwhile, in a small bowl, whisk remaining vinegar with mustard; set aside.
Pat liver dry; brush with oil and sprinkle with remaining salt and pepper.
Add to greased grill; close lid and grill for two minutes. DO NOT OVER COOK- no shoe leather please.
Turn and brush with half of the vinegar mixture; grill covered until glazed; browned on both sides and slightly pink inside, about 1 minute.
Serve with mushrooms and onions. Sprinkle with parsley