**Preheat the Oven:** Preheat your oven to 375°F (190°C). Lightly grease a 9x13-inch baking dish and set aside.
**Cook the Mushrooms:** Heat olive oil or butter in a large skillet over medium heat. Add the mushrooms and cook for 6-8 minutes, stirring occasionally, until they release their moisture and start to brown. Add garlic and thyme, cooking for an additional 1 minute. Season with salt and pepper.
**Make the Sauce:** In a mixing bowl, whisk together the heavy cream, sour cream, and Parmesan cheese. Stir in half of the mozzarella cheese.
**Assemble the Casserole:** Transfer the cooked mushrooms to the prepared baking dish. Pour the creamy sauce over the mushrooms, spreading it evenly. Top with the remaining mozzarella cheese and breadcrumbs.
**Bake:** Bake in the preheated oven for 20-25 minutes, or until the top is golden and bubbly.
**Garnish and Serve:** Remove from the oven and let cool slightly. Sprinkle with fresh parsley before serving.