Prepare the Mushrooms: Preheat the oven to 375°F (190°C). Remove the mushroom stems and scoop out a bit of the center to create room for the filling. Arrange the caps on a baking sheet lined with parchment paper. Bake for 10 minutes, then set aside to cool.
Caramelize the Onions: Heat olive oil and butter in a skillet over medium heat. Add the onions with a pinch of salt, stirring occasionally for 20-25 minutes until golden and caramelized. Add garlic and cook for an additional minute.
Prepare the Filling: In a bowl, mix the caramelized onions, panko breadcrumbs, Parmesan cheese, mozzarella, parsley, thyme, salt, and pepper. Taste and adjust seasoning as needed.
Stuff the Mushrooms: Fill each mushroom cap generously with the prepared mixture, pressing slightly to ensure the filling stays in place.
Bake to Perfection: Return the stuffed mushrooms to the oven and bake for 15-20 minutes, until the tops are golden and bubbly. Garnish with fresh parsley before serving.