Prepare the bread. Tear apart the bread into 1 1/2- to 2-inch pieces with your hands. Place the bread pieces on a large rimmed sheet pan and spread into an even layer. If you are making the stuffing the following day, allow the bread to sit uncovered at room temperature to dry out. If you are making the stuffing on the same day, preheat the oven to 300ºF/150ºC. Toast the bread pieces in the oven until completely dried out and lightly crisped, about 25 – 40 minutes. The exact amount of time will depend on the type of bread you’re using, as some breads have more moisture than others.
Prepare the flax egg. Combine 1 tablespoon flaxseed meal with 2 1/2 tablespoons water. Stir to combine and set aside for 15 minutes to thicken.
Preheat the oven to 350ºF/175ºC. Lightly grease a 9×13-inch rectangular baking dish (or other 3-quart baking dish with oil or vegan butter). If you want to make the stuffing “wreath” shape, use a 12-cup Bundt pan.
Heat a deep saucepan or Dutch oven over medium heat. Once hot, add the olive oil and vegan butter. Once melted and shimmering, add the wild mushrooms and button or cremini mushrooms. Cook until the mushrooms are lightly browned and most of the liquid has evaporated, stirring occasionally but not too often to allow them to brown, about 8-9 minutes. Add 1 teaspoon of the kosher salt and black pepper to taste and cook for 1 more minute.
Add the leeks, garlic, and chopped sage, rosemary, and thyme. Cook for 3-5 minutes, stirring occasionally to prevent burning, or until the leeks are starting to turn golden.
Pour in the white wine, and stir to deglaze the pan. Stir occasionally, until almost all of the wine evaporates, about 5-7 minutes. Add the miso paste and stir to incorporate into the veggies. Transfer the mushroom mixture into a very large bowl (the biggest bowl you have so you have enough room for mixing).
Add the flax egg, dry bread cubes, and chopped parsley to the mushroom mixture. Toss well to combine. Season with remaining ½ teaspoon kosher salt and black pepper to taste. Pour 1 cup of vegetable broth over the mixture to moisten, stirring to coat. Allow the stuffing to rest for 1-2 minutes to absorb the liquid. Then toss again until the liquid is evenly distributed onto each piece of bread. Add more broth as needed, depending on how dry your bread is.
If using vegan parmesan, sprinkle it over the stuffing and gently mix it in.
Transfer the stuffing to your prepared dish. Spray a piece of aluminum foil lightly with cooking spray or brush lightly with oil to prevent sticking. Cover the pan tightly with the foil bake until warmed through, about 25 minutes. Then increase the oven temperature to to 425°F/220°C and uncover the pan. Continue baking until golden-brown and crisp on top, about 15-20 minutes.
Allow to cool slightly before serving. If you are making the stuffing in a bundt pan, allow to cool for 20-25 minutes before inverting onto a serving dish.