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Bring a large covered pot of heavily salted water to a boil.
Meanwhile, in a 12-inch ovenproof skillet with high sides and a tight-fitting lid (or a Dutch oven), heat the oil over medium-high until shimmering. Add the squash and season with salt, cumin and red-pepper flakes. Cook, stirring every minute, until squash becomes browned in spots and feels just tender, 6 to 8 minutes.
Meanwhile, heat the oven to 400 degrees. Add the pasta to the boiling water and cook, uncovered, until not quite al dente, 3 to 4 minutes less than the package instructions. (It should be a little too firm to the bite.) Reserve 1 cup of the pasta water and drain. If the pasta is done before the squash, then stir in a drizzle of olive oil so that it doesn’t stick together.
When the squash is just tender, add the broth. Bring to an active simmer, cover and cook, stirring occasionally, until the squash is soft and easily mashable, 10 to 12 minutes. Turn off the heat, then use a potato masher or the back of a wooden spoon to crush about half of the butternut squash and leave the rest chunky. Season the squash to taste, keeping in mind that salty Parmesan will be added soon.
Add the cooked pasta to the skillet along with 1 cup reserved pasta water and 1/2 cup grated Parmesan, stirring vigorously to combine. Stir in the spinach one handful at a time until it shrinks down a little.
Sprinkle the top with the remaining 1/4 cup Parmesan, the mozzarella and jalapeño, then place in the oven, on a sheet pan if you are worried about dripping. Cook until the top is melted and browned in spots, 12 to 15 minutes. Drizzle with olive oil, top with parsley and serve.