1/4 cup (1/2 stick) unsalted butter
2 1/2 cups chopped onions
1/2 pound crimini (baby bella) mushrooms, sliced (about 3 cups)
2 pounds butternut squash, peeled, seeded, cut into 1/4-inch-thick slices (about 5 1/2 cups)
1 14-ounce can vegetable broth
4 tablespoons chopped fresh thyme, divided
4 tablespoons sliced fresh sage, divided
3 15-ounce containers whole-milk ricotta cheese
4 cups grated mozzarella cheese, divided
2 cups grated Parmesan cheese, divided
4 large eggs
Olive oil
1 9-ounce package no-boil lasagna noodles