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