Preheat the oven to 190°C (375°F) and lightly grease your baking dish.
Gently clean the mushroom caps with a damp cloth, then carefully remove the stems and set them aside for the filling.
Finely chop the mushroom stems and set aside. Mince the garlic cloves as well.
Heat a tablespoon of olive oil in a sauté pan over medium heat. Add the chopped stems and minced garlic, cooking for about 3 minutes until fragrant and slightly browned.
Transfer the sautéed mixture to a small bowl and stir in the chopped thyme, a pinch of salt and pepper, and half of the Parmesan cheese.
Stuff each mushroom cap generously with the filling mixture, pressing lightly to pack it in.
Place the stuffed mushrooms in the baking dish, then sprinkle the tops evenly with breadcrumbs, remaining Parmesan, and shredded mozzarella if using. Drizzle lightly with olive oil for extra crispiness.
Bake in the preheated oven for about 20 minutes, or until the tops are golden brown and bubbling.
Remove from the oven and let rest for 5 minutes. Garnish with fresh herbs or a squeeze of lemon for added brightness.