Preheat barbecue on medium. Combine thyme, olive oil, garlic and 1 teaspoon salt in a shallow bowl. Toss through mushrooms, packing thyme and garlic into mushroom caps. Cook mushroom, gill-side up, for 20 minutes.
Meanwhile, combine parmesan and cream in a small saucepan on low heat. Stir constantly until smooth.
Place mushrooms gill-side up on bread slices. Drizzle with parmesan cream and a little balsamic vinegar and serve with extra thyme sprigs and baby spinach.