In a bowl, combine cheese wedges, zucchini, chopped mushroom stems, onion, garlic, salt, and thyme. Add pepper, to taste. Mix until smooth. Set aside.
Lay a large piece of heavy-duty foil on a baking sheet. Lightly spray both sides of the mushroom caps with olive oil nonstick spray, and place next to each other on the foil with the rounded sides down. Spoon veggie-cheese mixture into the mushroom caps-there will be a lot, so pack it in!
Sprinkle with grated topping.
Place another large piece of foil over the caps. Fold together and seal all four edges of the two foil pieces, forming a well-sealed packet.
Place the baking sheet in the oven and bake for 23 to 25 minutes, until mushrooms are tender.
Allow the packet to cool for a few minutes, and then cut it to release steam before opening it entirely. (Careful-steam will be hot.)
If you like, season with additional salt and thyme, to taste. Enjoy!