1. Place mushrooms in a glass dish, top down.
2. Whisk together the vinegar, pineapple juice, and ginger. Drizzle the marinade over the mushrooms. Cover and marinate in the refrigerator for about 1 hour. Reserve the marinade for basting.
3. Heat a grill or broiler to high. Oil the rack. Grill or broil the mushrooms on medium heat, turning once, until tender (5 minutes per side). Baste with marinade to keep from drying out. Garnish with basil to serve.