Special equipment: kitchen twine
Preheat the oven to 425 degrees F.
Toss the squash with 2 tablespoons of the oil, 3/4 teaspoon salt and 1/4 teaspoon pepper. Spread on a baking sheet and roast until tender and golden brown in spots, about 40 minutes, rotating the baking sheet halfway through.
Meanwhile, combine the broth and 2 cups water in a saucepan and heat until just barely simmering. Tie together the sage with kitchen twine and add to the stock. Reserve for a later use.
Heat the remaining 2 tablespoons oil in a low-sided stockpot set over medium heat. Add the shallots and cook, stirring frequently with a wooden spoon, until translucent and soft, 6 to 8 minutes. Add the garlic and cook until softened, about 3 minutes. Add the oats, 1/2 teaspoon salt and 1/4 teaspoon pepper, and stir until coated. Add the wine and simmer, stirring constantly, until absorbed, about 1 minute. Add 3 cups of the warm broth to the oats, then simmer, stirring constantly, until most of the liquid is absorbed, 8 to 10 minutes. Add 2 more cups of the broth and simmer, stirring constantly, until the oats are tender but still have a slight bite, 10 to 12 minutes. (There will be a fair amount of liquid in the pot at this time, similar to the texture of a thin gravy. It will thicken once the cheese is added.) Remove from the heat and stir in half of the cheese and half of the parsley. Adjust the consistency of the risotto by adding more stock as needed. Divide the hot risotto among four warm serving bowls, and top each with some butternut squash. Sprinkle with the remaining cheese and parsley to serve.
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