Special equipment: a mandoline
Preheat the oven to 375 degrees F. Spray a 9-by-13-inch baking dish with cooking spray.
Poke the squash all around with a fork, then microwave it in 5-minute increments, rotating in between, until the outside is soft and yields a little when pressed with your fingers, about 15 minutes total. Be careful when rotating and removing the squash, as it will become very hot.
Meanwhile, combine the cream, butter, garlic, thyme, 1/2 cup of the Parmesan, 1 1/2 teaspoons salt and 1/4 teaspoon pepper in a large saucepan. Peel the potatoes, then slice them paper thin on a mandoline, placing them in the cream mixture as you slice. Bring the mixture to a simmer over medium-high heat, then remove from the heat.
When the squash is cool enough to handle, slice it in half, scoop out the seeds, then scrape out the flesh with a fork into a medium bowl. Discard the skin and seeds.
Layer the bottom of the prepared baking dish with 1/2 the potatoes. Top with 1/2 the squash, then the remaining potatoes, then the remaining squash. Pour all the liquid from the potato mixture into the baking dish, and press the vegetables down to submerge. Sprinkle the remaining 1/2 cup Parmesan on top. Put the baking dish on a rimmed baking sheet, and bake until golden brown and a knife slides easily into the potatoes, about 1 hour. Let cool for 15 minutes before serving.
The squash can also be roasted in the oven. Halve it lengthwise and scoop out the seeds, place it, cut-side down, on a parchment-lined baking sheet and roast in a 375-degree F oven for 1 hour or until tender.
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