Preheat oven to 400* F. Arrange the bread pieces in a single layer on a rimmed baking sheet. Bake until crisp and dried out but not brown (about 10 minutes).
In a large skillet, melt vegan butter over medium heat. Add celery, carrots, shallots (or onions; and chestnuts if using). Cook until vegetables are soft, (about 4 minutes). Add garlic, sage, and thyme, and cook another 1-2 minutes.
Add mushrooms and continue to cook until they lose their moisture.
Add wine (if using) and cook until evaporated (3-5 minutes).
Transfer cooked veggies to a very large bowl or pot. Add dried bread, parsley, and salt and pepper. Stir to combine.
Mix in half of the broth. Then continue to add the rest of the broth as needed, just until the stuffing is moistened but not too wet. There should not be any liquid pooled in the bottom of the bowl.
Butter the bottom of a casserole dish. Spoon mixture into casserole and cover with buttered aluminum foil. Bake for 30 minutes. Uncover the dish and bake for another 15 minutes or until golden.