Preheat the oven to 375°F.
Bring a large pot of water to a boil for the pasta.
In a large skillet, heat the oil, two turns of the pan, over medium-high. Season the chicken with salt and pepper, and add to the skillet, skin-side down. Cook until the skin is crispy and the chicken is golden, 4-5 minutes per side; transfer to a baking sheet (reserve the skillet) and bake until cooked through, about 5 minutes. Let the chicken rest while you make the sauce.
Add the vegetables and garlic to the skillet; season and cook, stirring often, until soft, 8-10 minutes. Add the stock and simmer, using a fork to coarsely mash the vegetables. Add the parsley.
Remove the skin and bones from the chicken and thinly slice. Salt the boiling water, add the pasta and cook to al dente. Add half a mug of the starchy cooking water to the sauce.
Drain the pasta and toss with sauce, a handful of cheese and a fat drizzle of EVOO. Serve in shallow bowls, passing more cheese at the table.