Preheat the oven to 325 degrees F. Brush a baking sheet with 1 teaspoon of the oil.
Finely chop the mushroom stems and set aside. Place the mushroom caps on the prepared baking sheet and bake for 12 minutes.
Meanwhile, heat the remaining 1 teaspoon oil in a medium skillet over medium heat. Add the mushroom stems, spinach, onion, and garlic and saute until the spinach wilts, about 6 minutes. Add the tomatoes and cook for 2 minutes more. Turn off the heat and stir in the parsley, salt, pepper and 2 tablespoons each of the pecorino and panko.
Preheat the broiler. Divide the spinach mixture evenly among the mushroom caps. Sprinkle each with the remaining 1 tablespoon pecorino and panko, then broil until golden brown, 2 to 3 minutes. Serve warm.