In a large bowl, blend miso, sake, ginger, honey, lemon juice, and vinegar.
Add mushrooms; stir to coat with marinade, then cover and refrigerate for at least 6 hours or until next day.
Drain marinade into a small pan and set aside.
Thread an equal number of mushrooms on each of 8 small skewers, then place them on a lightlygreased grill 4 to 6 inches above a solid bed of low glowing coals.
Cook, turning once or twice and brushing occasionally with reserved marinade, until browned on all sides (5 to 10 minutes).
Or, to cook indoors, place skewers on a broiler pan and broil about 6 inches from heat, turning as needed and brushing occasionally with reserved marinade, for 6 to 8 minutes.
Remove mushrooms from skewers.
Heat remaining marinade and serve it in a small bowl, topped with green onions, as a dipping sauce.
Serve mushrooms with wooden picks for spearing.