1. In a 10- to 12-inch frying pan over high heat, stir mushrooms and shallots in butter until mushrooms are browned, about 10 minutes. Add thyme, marsala, and cream. Stirring often, boil until mixture is pale gold color and there is no free-flowing liquid, 6 to 7 minutes. Let cool to room temperature.
2. Rinse sole, pat dry, and lay fillets flat. Spoon mushroom mixture equally onto fillets and spread to coat evenly. Roll fillets from narrow end and set seam down on a sheet of foil (about 9 by 13 in.).
3. Set foil with fish on a barbecue grill over a solid bed of hot coals or high heat on a gas grill (you can hold your hand at grill level only 2 to 3 seconds). Cover with lid (open vents for charcoal). Cook until sole is opaque but moist-looking in thickest part (cut to test), about 8 minutes. The last 2 minutes, sprinkle cheese evenly on fish.
4. Slide a rimless baking sheet under foil and lift from grill. With a wide spatula, transfer sole to plates; scrape any drippings from foil and spoon onto fish. Garnish with thyme sprigs. Add salt to taste.