Preheat the oven to 210 degrees. Peel the butternut squash, remove the seeds, and cut the flesh into three-centimeter chunks. Toss them with the olive oil, salt, and black pepper, and spread them out well on a baking sheet. This space is not a minor detail: if the chunks lie against each other, the released moisture gets trapped between the pieces and the pumpkin steams into a watery mass instead of browning. Roast for twenty to twenty-five minutes until the edges caramelize darkly.
Meanwhile, dry-toast the walnuts in a frying pan until they smell warm, and chop them coarsely after cooling.
Bring a shallow pan of water with the white wine vinegar to a gentle simmer, so that only small bubbles rise from the bottom. Crack the eggs one by one into a cup and gently slide them into the water. Poach them for three minutes and remove them with a slotted spoon.
Melt the butter in a small pan until it foams and smells slightly nutty, and fry the sage leaves in it for half a minute until crispy.
Spoon the pumpkin onto a plate, place the eggs on top, and spoon the sage butter over everything. Shave the pecorino on top, sprinkle with the walnuts and chili flakes, and grind some black pepper over it.