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1. Make the filling: Preheat oven to 400 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 and puree in a food processor until smooth 2. Melt butter in a small saucepan over medium heat. cook until browned ~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.
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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 until thickened ~ 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.
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6. Reduce 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 8. Top pasta with 3/4 cup bechamel and 2 tablespoons Parm Layer in same order until you reach the top of the dish, ending with pasta - 10 layers) Fold in the overhang from the first layer of pasta to create a package. 9. Bake covered w/ parchment-lined foil for 20 minutes Uncover sprinkle with parm bake to browned ~35 min stand
