For the chile sauce: Put the tomato, chiles, and salt into the bowl of a food processor and blitz until minced, scraping down the sides of the bowl. Add the vinegar and olive oil and pulse for a few seconds. Transfer to a small bowl and set aside.
Preheat the oven to 475 F. Line a baking sheet with parchment paper, and line a plate with a double layer of paper towel.
Put the squash, onion, 3 tablespoons of the olive oil, 3/4 tsp salt, and black pepper into a medium bowl and mix until well combined. Transfer to the prepared baking sheet and roast for 30 minutes, stirring once or twice, until soft and charred.
Meanwhile, put the garlic, and 2 tablespoons of olive oil into a small frying pan and place on medium heat. Cook gently, stirring occasionally, until the garlic becomes deeply golden and crispy, about 12 minutes. Reserving the oil, use a slotted spoon to transfer the garlic to the prepared plate.
Cook the pasta in plenty of salted boiling water until almost al dente, about 7 minutes. Reserving 3/4 cups of cooking liquid, drain the pasta well.
Put the yogurt, remaining 1 tablespoon of olive oil, cumin, egg yolk, cornstarch, and 1/4 teaspoon of salt into a blender and blitz until smooth, about 1 minute. Pour the mixture into a large saute pan on a medium heat. Cook, stirring continuously, until thickened and just beginning to bubble, about 15 minutes. Add the pasta, reserved pasta water, chopped cilantro, and half the squash mixture and cook for another 4 minutes. Transfer to a large serving platter and top with the remaining squash mixture. Drizzle with one-third of the chile sauce and top with fried garlic, cilantro and reserved garlic oil. Serve with shredded pecorino cheese.