1. Preheat oven to 350°.
2. Heat a large skillet over medium-high heat. Add oil; swirl to coat. Add green onions; sauté 30 seconds. Add 6 cups squash; sauté 2 minutes. Reduce heat to medium-low, and stir in 1/4 teaspoon salt. Cook, covered, 15 minutes or until tender. Increase heat to medium-high. Cook, uncovered, 2 minutes or until liquid evaporates, stirring frequently. Place squash mixture in a large bowl; mash with a potato masher or fork until smooth.
3. Cook bacon in a large nonstick skillet over medium-high heat until crisp. Remove bacon from pan, reserving drippings in pan; crumble bacon. Add chanterelles, shiitakes, and sage to drippings; sauté 8 minutes or until mushrooms are browned. Add stock; cook 3 minutes or until liquid almost evaporates. Add mushroom mixture, remaining 2 cups diced squash, remaining 1/4 teaspoon salt, 1 ounce Parmigiano-Reggiano cheese, ricotta cheese, and black pepper to mashed squash mixture, stirring to combine. Spoon squash mixture into a broiler-safe 11 x 7-inch glass or ceramic baking dish coated with cooking spray. Cover with foil; bake at 350° for 1 hour. Remove pan from oven; discard foil.
4. Preheat broiler to high.
5. Combine bacon, remaining 1 ounce Parmigiano-Reggiano, and oyster mushrooms in a bowl; sprinkle over gratin. Broil 6 minutes or until lightly browned. Let stand 5 minutes before serving.