Without a grill, the mushrooms can be cooked in a grill pan on the stove or under the broiler. They are meant to be done in advance and kept warm, leaving any last-minute preparation for finishing the rice.
Mix 3 tablespoons oil, vinegar, 1/2 cup wine, half the garlic, 4 branches thyme, salt and pepper in a dish to hold mushrooms in a single layer. Chop stems and set aside. Place caps in dish and coat with marinade. Marinate 2 hours, turning from time to time.
Heat grill or broiler. Slowly grill or broil caps, turning and basting with marinade until tender, 10 minutes or so. Wrap in foil and set aside. Place stock in a pot over low heat.
Heat remaining oil in a shallow 3- to 4-quart saucepan. Add onion, chopped stems, leaves from 4 branches thyme and remaining garlic, and sauté over medium heat until onion is soft. Stir in rice and cook two minutes. Stir in remaining wine and cook until nearly gone, about 2 minutes. Add stock, about 1/2 cup at a time, stirring and adding more as liquid is absorbed, until rice is al dente. Open wrapped mushrooms and pour any liquid into rice. Stir for a minute. Season with salt and pepper.
Divide risotto among 4 warm soup plates or dinner plates. Place a mushroom on each, garnish with a branch of thyme, and serve with grated cheese alongside.