Mix all marinade ingredients: balsamic vinegar, wine, olive oil, spices, minced garlic.
Brush the mushroom caps. Remove stems carefully by pulling and twisting or slicing out. (You can use the stems in another dish if you'd like).
Place caps, stem-side up, in a shallow dish. Pour marinade over them, filling the cavity where the stems were and pouring marinade around the mushrooms. Gently tip the caps to allow excess liquid to run out of the cavity but leave some in for moisture. Cover with plastic wrap and refrigerator for 2 hours.
Remove mushrooms from the refrigerator and turn in the dish to ensure the entire cap is marinaded. Preheat George Foreman grill or grill for 15 minutes until hot.
Remove mushrooms from marinade. Place on grill. Grill 6 minutes, total, in a George Foreman grill, or if using conventional grill, 3-5 minutes on each side until done. Remove, serve.