Place 2 tablespoons of the olive oil into an 8-quart stockpot over high heat until the oil shimmers. Add the mushrooms and saute until browned, about 5 minutes. Remove the mushrooms from the pan and set aside.
Decrease the heat to low, add the remaining 2 tablespoons olive oil and the carrots, celery, onions and salt. Cook, stirring occasionally, until softened, about 30 minutes.
Add the tomatoes, sage, rosemary and garlic, and cook for 5 minutes, stirring to break up the tomatoes. Add the water, soy sauce and Parmesan rind, increase the heat to high and bring to a boil, about 15 minutes. Decrease the heat to low, add the squash and kale, cover and cook until tender, 30 to 35 minutes.
Return the mushrooms to the pot along with the beans and red wine vinegar and cook until all is heated through, about 15 minutes. Remove the cheese rind and serve warm with shaved Parmesan.