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Peel the squash, remove the seeds and cut into 1-inch chunks.
Preheat the oven to 400 degrees.
Combine the squash and onion on a large rimmed baking sheet. Toss with the olive oil, 1 teaspoon salt and 10 or 12 grinds of black pepper. Cover the pan with aluminum foil and bake 25 minutes. Remove the foil and stir the squash around with a spatula. Roast 15 minutes more, or until the squash is tender and lightly browned.
Pour the cream into a saucepan. Drop in the garlic cloves and thyme sprigs and bring to a simmer — lower the heat and watch that it doesn't boil over. Simmer 20 minutes, until the cream is slightly reduced and the garlic is soft. Remove from the heat and steep 10 minutes, then remove the garlic and thyme. Stir in 1/2 teaspoon salt and 1/2 cup of the cheese. Transfer the sauce to a serving bowl with the butter and keep warm.
Bring a 6 quart pot of water to a boil with 2 tablespoons kosher salt. Cook the pasta until al dente. Drain and add to the bowl with the sauce.
Add the squash and baby greens to the bowl and gently stir everything together until evenly coated with the sauce. Serve with the remaining cheese and chopped thyme sprinkled over the top.