Season the chicken with salt and pepper on both sides.
Heat the EVOO, two turns of the pan, in a Dutch oven over medium-high heat. Brown the chicken, skin-side down, first in 2 batches, until crispy; remove to a plate. Add the pancetta, guanciale or bacon and crisp; drain off some of the fat if there is more than 5-6 tablespoons. Add the mushrooms and brown for 7-8 minutes. Add the Fresnos, carrots, garlic, onion, bay leaf, rosemary, thyme, salt and pepper. Partially cover the pan and cook for 7-8 minutes to soften the vegetables. Stir in the tomato paste; cook for 1-2 minutes. Add the wine and reduce by half. Add the tomatoes and basil; break up the tomatoes and bring to a simmer. Add the chicken back to the pot and simmer for 35-40 minutes, uncovered, to thicken. Stir in the parsley near serving time.
Bring the pasta water to a full rolling boil. Season the pasta water with salt and cook the pasta to al dente.
Arrange the chicken pieces on a platter and cover with some of the sauce.
Drain the pasta and add back to hot pot and toss with the butter, half the remaining sauce and the cheese. Transfer the pasta to a platter or bowl and top with additional sauce.