In a small bowl cover the porcini with 1 cup boiling water; soak until softened about 30 minutes. Finely chop the porcini. Let the sand settle on the bottom, carefully pour off and reserve ½ cup of the soaking liquid.
In a large cast iron pot, heat 2 Tb of the olive oil. Add onion and carrots and cook over medium-low heat until lightly browned, about 8 minutes.
Add eggplant and 2 Tb olive oil. Raise the heat to medium and cook stirring occasionally until softened, about 8 minutes.
Stir in the cremini, shitake, chopped porcini and remaining 2 Tb olive oil. Season with salt and pepper
Cook stirring occasionally until the mushrooms begin to brown 8-10 minutes.
Stir in the tomato paste, miso and garlic. Stir to dissolve the miso and tomato paste. Cook for 1 minute.
Add the tomatoes and their juices, thyme, sugar and reserved mushroom liquid and bring to a simmer
Cover and cook over low heat, stirring occasionally, until the sauce is very thick about 1 hour.
Discard the thyme sprig and season with salt and pepper.
In a pot of salted boiling water, cook the spaghetti until al dente. Drain reserving ½ cup of the pasta water.
Add pasta, pasta water and parsley to the sauce, toss to coat. Serve in bowls topped with cheese.