Preheat the oven to 375 degrees .
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; add to the skillet, skin side down. Cook until the skin is crispy and the chicken is golden, 4 to 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 to 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; 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; toss with the sauce, the chicken, a handful of cheese and a fat drizzle of EVOO. Serve in shallow bowls, passing more cheese.