Melt butter in large skillet over medium-high heat.
Add onions; sauté until soft, about 8 minutes. Increase heat to high; add mushrooms and cook until tender, stirring constantly, about 3 minutes. Season with salt and pepper.
Transfer mushroom mixture to bowl; set aside.
Add squash, broth, 3 tablespoons thyme, and 3 tablespoons sage to same skillet. Cover and simmer over medium heat until squash is just tender, about 6 minutes. Uncover and cook until squash is very soft but still retains shape, about 5 minutes. Season with salt and pepper.
Mix ricotta, 2 cups mozzarella cheese, 1 1/2 cups Parmesan cheese, and remaining 1 tablespoon thyme and 1 tablespoon sage in large bowl. Season to taste with salt and pepper; mix in eggs.
Brush 13x9x2-inch glass or ceramic baking dish with oil.
Spread 1 cup ricotta mixture over bottom. Arrange 3 noodles on top.
Sprinkle with 1/2 cup mushrooms and 1 cup mozzarella. Top with 3 noodles, then 1 3/4 cups ricotta mixture, half of remaining squash, 1/2 cup mushrooms, and remaining 1 cup mozzarella. Repeat with noodles, 1 3/4 cups ricotta mixture, remaining squash, and remaining mushrooms. Top with 3 noodles.
Spread remaining ricotta mixture over; sprinkle with remaining Parmesan. Cover with oiled foil.
Preheat oven to 350°F.
Bake lasagna, covered, 35 minutes. Uncover; bake until heated through, about 25 minutes longer.
Let stand 10 minutes before serving. (The lasagna can be assembled one day ahead and refrigerated.)
