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1
Heat the oven to 400 degrees F and arrange a rack in the top third.
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2
Prepare the squash: Peel off the skin with a vegetable peeler.
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3
Trim the top and bottom.
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4
Cut the neck from the bulb of the squash.
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5
Halve each piece lengthwise and scrape out the seeds.
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6
Cut the squash into 1/4-inch cubes and set aside.
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7
(You will need about 5 cups.
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8
Save any remaining squash for another use.)
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9
Heat olive oil in a large frying pan over high heat.
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10
When it begins to smoke, add mushrooms and season with salt and freshly ground black pepper.
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11
Cook, stirring occasionally, until tender and crispy at the edges, about 4 minutes.
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12
Stir in squash and cook, stirring frequently, until squash is browned.
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13
Add garlic and cook until just fragrant, about 1 minute more.
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14
Add water and stir, scraping up any browned bits that have accumulated on the bottom of the pan.
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15
Taste and adjust seasoning as needed.
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16
Remove the pan from heat and set aside.
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17
Spread 1/4 cup of the bechamel sauce over the bottom of a 13-by-9-inch baking dish.
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18
Cover with a layer of lasagna noodles, breaking the noodles as necessary to fit them in an even layer.
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19
Evenly spread half of the vegetable mixture over the pasta.
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20
Pour 1/2 cup of the bechamel sauce evenly over the vegetables.
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21
Dot half of the chestnut puree over top.
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22
Cover the vegetables with a layer of grated fontina cheese and a sprinkling of Parmigiano-Reggiano (use about a third of each).
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23
Repeat with another layer of noodles, the remaining squash mixture, 1/2 cup bechamel, the remaining chestnut puree, another third of the fontina, all of the Gorgonzola, and about another third of the Parmigiano-Reggiano.
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24
Finish with a last layer of noodles and the remaining bechamel sauce, fontina cheese, and Parmigiano-Reggiano.
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25
Cover the baking dish well with aluminum foil.
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26
(You can prepare this recipe ahead and refrigerate for up to 1 day; allow to come to room temperature before baking.)
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27
Bake for 30 minutes, then remove the foil and continue baking until the top is brown and the noodles are completely tender, about 10 minutes more.
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28
Allow the lasagna to stand for 10 to 15 minutes before serving.