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1
Heat the oven to 450 degrees F and arrange a rack in the middle.
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2
Heat 2 tablespoons of the oil in a large saucepan over medium heat until shimmering.
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3
Add 1/3 cup of the shallots, the rosemary, and thyme, season with salt, and cook, stirring occasionally, until the shallots are softened, about 4 minutes.
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4
Increase the heat to high, add the measured water, milk, and measured salt, stir to combine, and bring to a simmer.
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5
Reduce the heat to low and, while whisking constantly, slowly pour in the polenta in a thin, steady stream until all of it is incorporated and there are no lumps.
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6
Cook, whisking or stirring with a wooden spoon occasionally, until the polenta pulls away from the sides of the pan and the grains have softened, about 35 to 40 minutes.
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7
(If the polenta starts to get too thick before its done, add water 1/4 cup at a time as necessary.)
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8
Meanwhile, prepare the squash.
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9
Using a vegetable peeler, peel the squash, then trim the ends.
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10
Cut it in half lengthwise and remove the seeds.
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11
Cut each half crosswise into 1/4-inch-thick slices.
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12
Place the slices in a large bowl, add 2 tablespoons of the oil, season generously with salt and pepper, and toss with your hands to evenly coat.
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13
Spread the squash in an even layer on a baking sheet.
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14
Roast until knife tender, about 20 minutes.
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15
Wipe out the bowl and set it aside (no need to wash it).
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16
When the squash is ready, transfer it to a wire rack to cool.
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17
Reduce the oven temperature to 375 degrees F.
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18
When the polenta is ready, stir in the Gruyere cheese until its melted and evenly incorporated.
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19
Taste and season with salt as needed.
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20
Transfer the polenta mixture to a 17-by-12-inch rimmed baking sheet.
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21
Using a rubber spatula, spread it into an even layer; set aside.
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22
Heat the remaining 2 tablespoons of oil in a large frying pan over medium heat until shimmering.
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23
Add the remaining 1/3 cup of shallots, season with salt and pepper, and cook, stirring occasionally, until softened, about 4 minutes.
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24
Increase the heat to medium high, add the wine, and cook until it has almost evaporated, about 3 to 4 minutes.
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25
Add the chard, season with salt and pepper, and cook, tossing constantly with tongs, until its completely wilted and tender and the liquid has almost evaporated, about 3 to 5 minutes.
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26
Transfer the mixture to a colander in the sink and let it sit, stirring occasionally, until its cool enough to handle, about 10 minutes.
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27
When the chard mixture has cooled, squeeze large handfuls to release any excess liquid and transfer to a cutting board.
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28
Finely chop and place in the reserved bowl.
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29
Add the ricotta and 3/4 cup of the Asiago and stir to combine.
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30
Taste and season with salt and pepper as needed.
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31
Add the egg and stir until evenly combined.
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32
Assemble the lasagna: Cut the polenta in half crosswise (you should have 2 pieces that are approximately 8 by 6 inches each).
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33
Using a flat spatula, transfer 1 of the halves in large pieces to a 13-by-9-inch baking dish and, using your hands, press into an even layer to completely cover the bottom of the dish.
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34
Using a rubber spatula, evenly dollop half of the ricotta mixture over the polenta and spread it into an even layer.
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35
Evenly shingle half of the squash pieces over the ricotta mixture.
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36
Repeat the layers once more with the remaining halves of the polenta, ricotta mixture, and squash.
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37
Sprinkle with the remaining 1/4 cup of Asiago cheese.
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38
Bake until the cheese is melted and the top is golden brown, about 30 to 40 minutes.
