How to Make It
In a small bowl, cover the porcini with 1 cup of boiling water; soak until softened, about 30 minutes. Finely chop the porcini, discarding any tough bits. Pour off and reserve 1/2 cup of the soaking liquid.
In a large enameled cast-iron casserole, heat 2 tablespoons of the oil. Add the onion and carrots and cook over moderate heat until light golden, about 8 minutes. Add the eggplant and 2 tablespoons of the oil and cook, stirring occasionally, until softened, 8 minutes. Stir in the cremini, shiitake, chopped porcini and the remaining 2 tablespoons of oil and season with salt and pepper. Cook, stirring occasionally, until the mushrooms are golden, 8 to 10 minutes. Stir in the garlic, tomato paste and miso and cook for 2 minutes. Add the chunk of cheese, the tomatoes and their juices, the thyme, sugar and reserved mushroom soaking liquid and bring to a simmer.
Cover and cook over low heat, stirring occasionally, until the sauce is very thick, about 1 hour and 30 minutes. Discard the thyme sprig; season the sauce with salt and pepper.
In a pot of salted boiling water, cook the spaghetti until al dente. Drain, reserving 1/4 cup of the pasta water.
Add the pasta, pasta water and parsley to the sauce; toss to coat. Serve in bowls, topped with grated cheese.