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.