1 1/2-ounce package dried porcini mushrooms, rinsed
3 tablespoons unsalted butter
3 tablespoons extra-virgin olive oil
1 pound button mushrooms, trimmed and quartered
1 pound shiitake mushrooms, stems removed, caps quartered
Kosher salt
1 small onion, chopped
2 cloves garlic, minced
1/4 teaspoon red pepper flakes
3 tablespoons tomato paste
1 15-ounce can whole San Marzano tomatoes, crushed by hand
1/2 cup chopped fresh parsley, plus more for topping
2 1/2 cups instant polenta
Kosher salt
1 cup grated parmesan cheese (about 4 ounces)
2 large eggs, lightly beaten
1/4 teaspoon freshly grated nutmeg
Extra-virgin olive oil, for the baking dish
3 cups grated part-skim mozzarella cheese