1. Make the filling: Preheat oven to 400 degrees. Drizzle squash halves with oil, and season with salt and pepper. Roast squashes, cut sides down, on a rimmed baking sheet until tender and browned, about 1 hour. Let cool. Scoop flesh from skins, and puree in a food processor until smooth.

2. Melt butter in a small saucepan over medium heat. Continue to cook until browned and fragrant, about 3 minutes. Transfer to a bowl.

3. Combine 3 cups squash puree, 1 cup Parmesan, 2 tablespoons plus 1 1/2 teaspoons amaretti cookies, sage, browned butter, 1 teaspoon salt, and nutmeg in a bowl. Adjust seasoning as desired with more Parmesan, amaretti cookies, sage, salt, and nutmeg.

4. Make the bechamel: Melt butter in a medium saucepan over medium heat. Whisk in flour. Reduce heat to low, and cook, whisking often, for 3 minutes (do not let flour brown).

5. Meanwhile, bring milk to a gentle simmer in another small saucepan over low heat. Gradually whisk hot milk into the roux, whisking constantly to prevent lumps from forming. Season with 1/4 teaspoon salt and the nutmeg. Raise heat to medium, and bring to a boil, whisking constantly. Cook, whisking, until thickened, about 10 minutes. Reduce heat to low, and cook until the raw flour taste is gone, 5 to 10 minutes more. Season with 3/4 teaspoon salt.

6. Reduce oven temperature to 375 degrees. Assemble the lasagna: Coat bottom and sides of a 9-by-13-inch baking dish with a thin layer of bechamel (about 1/2 cup). Arrange two 8-by-21-inch sheets cooked pasta on top, barely overlapping, to cover bottom and sides of dish. (Edges of pasta should hang over sides by about 2 inches.)

7. Top with 3/4 cup squash filling, and sprinkle with 1 tablespoon amaretti cookies. Top with 1 sheet of pasta (fill in any gaps with pieces of another sheet if needed), trimming edges as needed so that pasta fits neatly in dish.

8. Top pasta with 3/4 cup bechamel and 2 tablespoons Parmesan. Continue layering in same order (pasta, squash and amaretti, pasta, bechamel and Parmesan) until you reach the top of the dish, ending with 1 sheet of pasta (you may need to fill in gaps with pieces of another sheet; you should have 10 layers of pasta). Fold in the overhang from the first layer of pasta to create a package. Trim pasta sheets as needed to prevent too much overlap, then spread remaining bechamel over top.

9. Bake, covered with parchment-lined foil, for 20 minutes. Uncover, and sprinkle with remaining Parmesan. Bake until top is browned, about 35 minutes more. Let cool slightly before serving.