Preheat broiler. Lightly brush eggplant slices with 1 tablespoon of the oil and broil 4 to 5 minutes, until lightly browned. Set aside.
Preheat oven to 190C/375F. Pierce squash with a fork and bake for 45 to 50 minutes, until softened.
Meanwhile, in a large skillet over medium heat, saute onion inremaining oil until lightly browned. Stir in mushrooms, garlic, rosemary, basil, oregano, and chilli flakes, and saute about 5 minutes, until mushrooms have softened. Stir in tomatoes, wine, miso, salt, and pepper. Bring to a boil, then turn off heat.
In a 2.5 - 3 liters (quarts) casserole dish, arrange 1/3 of the eggplant slices to cover bottom. Top with 1/3 of tomato-mushroom sauce. Repeat twice more with remaining eggplant and sauce. Cover and bake at 190C/375F for about 25 minutes, until hot and bubbling.
Remove casserole and squash from oven. Slice squash in half and remove seeds. With a fork, scrape strands from squash and arrange over top of casserole.