Meaty walnuts, fresh herbs, and 2 kinds of mushrooms give this vegetarian “bolognese” its satisfying flavor and texture.
Heat a large pot of water to a boil for pasta.
Toast the walnuts in oven on small tray, moderate heat, or toast them in a small skillet until brown and fragrant. Chop.
Lay out a kitchen towel. Peel off half the skin of the eggplant and trim the top and bottom. Cut eggplant into thin steaks or planks, then chop into small dice and place on towel. Season with salt and let drain 20-30 minutes, then roll eggplant up in towel to press out excess liquids.
Place dried mushrooms and stock in small pot, then bring to simmer and reconstitute the mushrooms (plump them up).
Heat a deep skillet or Dutch oven over medium-high heat with EVOO, 3 turns of the pan. Add chopped mushrooms and brown well, then add the carrot, celery, onion, garlic, rosemary, sage, salt, and pepper and cook partially covered for 5 minutes to soften, stirring occasionally. Add eggplant and cook 7-8 minutes more again, partially covered. Add tomato paste and stir a minute, add chopped nuts, then the vermouth or white wine and let it absorb. Remove the dried mushrooms and finely chop them, then add to sauce along with all but the last few tablespoons of stock, where grit might have settled. Add milk to sauce and reduce heat to low and simmer gently 30 minutes to thicken and develop.
Cook pasta 1-2 minutes less than directions in salted water. Reserve 1 cup cooking water, then drain pasta and toss with it and the sauce and the cheese, if using. Serve in shallow bowls.