Preheat the oven to 350°/175° C.

Warm the large nonstick skillet over medium-high heat. Add the olive oil and swirl to coat. Sauté the green onions for 30 seconds. Add 6 cups of the squash and sauté for 2 minutes. Reduce the heat to medium-low, and then stir in ¼ teaspoon kosher salt. Cook, covered, until tender, about 15 minutes. Remove the lid and increase the heat to medium-high. Continue cooking until any liquid in the skillet evaporates, stirring frequently, about 2 more minutes. Transfer the squash mixture to the large bowl, and then mash with a potato masher or fork until smooth. Wipe the skillet clean.

Cook the bacon in the large nonstick skillet over medium-high heat just until crisp. Remove the bacon from the skillet, reserve the drippings, and crumble the bacon. Add the chanterelles, shiitakes, and sage to the skillet, and sauté until the mushrooms are browned, about 8 minutes. Add the beef stock and cook until the liquid almost evaporates, about 3 minutes.

Transfer the mushroom mixture to the mashed squash, and add the remaining 2 cups diced squash, remaining ¼ teaspoon salt, 1 ounce Parmigiano-­Reggiano cheese, ricotta cheese, and black pepper. Stir it all to combine. Spoon the squash mixture into the prepared baking dish. Cover with foil and bake in the preheated oven for 1 hour. Remove the pan from the oven and discard the foil.

Preheat the broiler to high. Sprinkle the crumbled bacon, remaining 1 ounce Parmigiano-Reggiano, and oyster mushrooms over the cooked squash. Broil until lightly browned, about 6 minutes. Remove from the oven and let stand 5 minutes before serving.