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1. Make filling: preheat oven to 400. Drizzle squash halves w/ oil, season w/ S+P. Roast squah, cut side down, on rimmed baking hseet until tender+browned, ~1h. Let cool Scoop flesh from skins, puree in food processor until smooth.
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2. Melt butter in small saucepan, med heat. Continue to cook until browned+fragrant, ~3min. Transfer to a bowl.
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3. Combine 3 cups squash puree, 1 cup Parm, 2 tbsp plus 1 1/2 tsp amaretti cookies, sage, browned butter, 1 tsp salt, nutmeg in a bowl.
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4. Make bechamel: melt butter in med saucepan, med heat. Whisk in flour. Reduce to low + cook, whisking often, 3 min (do not let flour burn).
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5. Meanwhile, bring milk to gentle simmer in another small saucepan, low heat. Gradually whisk hot milk into roux, whisking constantly to prevent lumps from forming. Season w/ 1/4 tsp salt + nutmeg. Raise heat to med + bring to a boil, whisking constantly. Cook, whisking, until thickened, ~10min. Reduce heat to low, cook until raw flour taste gone, 5-10 min more. Season w/ 3/4 tsp salt.
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6. Reduce oven to 375. Assemble lasagna: coat bottom+sides of 9x13 baking dish w/ thin layer of bechamel (~1/2 cup). Arrange two 8x21 sheets cooked pasta on top, barely overlapping, to cover bottom+sides. Edges of pasta should hang over sides by ~2in
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7. Top w/ 3/4 cup squash filling, sprinkle w/ 1tbsp amaretti cookies. Top w/ 1 sheet of pasta (fill in gaps w/ pieces of another sheet), trimming edges so that pasta fits in dish
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8. Top pasta w/ 3/4 cup bechamel + 2tbsp Parm. Continue layering: pasta, squash+amaretti, pasta, bechamel+Parm, until top of dish, ending w/ 1 pasta sheet (may need to fill in gaps w/ pieces of another sheet; should have 10 layers of pasta). Fold in overhang from first pasta layer to create a package. Trim pasta sheets to prevent too much overlap, then spread remaining bechamel over top.
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9. Bake, covered, w/ parchment-lined foil, for 20 min. Uncover, sprinkle w/ Parm. Bake until top browned, ~35min. Let cool slightly before serving.
