Mushroom-Polenta Lasagna

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For the sauce: 

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

For the lasagna:

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