Adjust oven rack to middle position and preheat oven to 425°F (220ºC). Brush bottom of a large (10- or 12-inch) cast iron skillet with 1 teaspoon oil; set aside. In a large bowl, mix the sausage, wine, garlic, thyme, oregano, salt, red pepper (if using), 2 tablespoons Parmigiano-Reggiano, and remaining 1 teaspoon of the oil until well combined.
Spoon about 2 teaspoons sausage mixture into each mushroom cap; place mushrooms in skillet in even layer, stuffed side up. Sprinkle filled mushrooms evenly with panko and remaining 1 tablespoon Parmigiano-Reggiano.
Bake until topping is golden brown, mushrooms are tender, and a thermometer inserted into filling registers at least 160°F (70℃), 15 to 20 minutes. Let cool for 5 minutes before transferring to a large serving plate. Garnish with parsley. Serve warm.
