Combine mushroom caps, olive oil, wine and seasoning in a resealable plastic bag. Marinate 30 minutes.
Preheat grill to medium or, if using charcoal, have coals at medium heat. Remove mushrooms from marinade; discard marinade.
Place mushroom caps stem side down (or use grill pan). Grill mushrooms 5 minutes; turn. Grill 3 to 4 minutes until mushrooms are almost cooked through.
Sprinkle Monterey Jack cheese evenly on mushrooms. Top with grated Parmesan. Continue grilling an additional 2 to 3 minutes. Remove from grill; let rest 3 minutes before serving.
