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1
Preheat the oven to 425 and butter a 9-by-13-inch baking dish.
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2
On a rimmed baking sheet, toss the squash with 2 tablespoons of the olive oil and season with salt and pepper.
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3
Bake for about 25 minutes, tossing once, until the squash is just tender.
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4
Reduce the oven temperature to 325.
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5
Meanwhile, in a large skillet, heat 2 tablespoons of the oil.
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6
Add the sliced onions, season with salt and cook over moderately low heat, stirring occasionally, until golden, about 25 minutes.
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7
Scrape the onions into a bowl.
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8
In the same skillet, heat the remaining 1 tablespoon of olive oil until shimmering.
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9
Add the kale, garlic, crushed red pepper and 1 teaspoon of the thyme and season with salt.
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10
Cook over moderately high heat, tossing, until the kale is wilted and just tender, about 5 minutes.
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11
Scrape the kale into the bowl with the cooked onions.
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12
In a medium saucepan, melt the 2 1/2 tablespoons of butter.
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13
Add the chopped onion and the remaining 1 teaspoon of thyme and cook over moderately low heat, stirring, until softened, 5 minutes.
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14
Add the flour and cook over moderate heat, whisking constantly, until a light golden paste forms, 3 minutes.
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15
Whisk in 1 cup of the milk and cook, whisking, until very thick and no floury taste remains, 8 to 10 minutes.
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16
Remove from the heat and whisk in the cream, creme fraiche, sugar, 2 teaspoons of salt, 1/2 teaspoon of pepper and the remaining 1 1/2 cups of milk.
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17
Let the bechamel cool.
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18
Beat the eggs into the cooled bechamel in the saucepan.
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19
Pour into a bowl, add the bread and the vegetables and mix well.
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20
Pour the strata mixture into the prepared baking dish and let stand for 30 minutes, pressing down the bread occasionally.
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21
Bake the strata for 55 minutes to 1 hour, until almost set.
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22
Increase the oven temperature to 475.
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23
Sprinkle the Parmigiano on the strata and bake for about 10 minutes more, until the top is lightly browned.
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24
Let the strata stand for 15 minutes before serving.
