Place the chicken thighs and breasts in a medium pot with the carrots, celery, quartered onion and bay leaves. Sprinkle with salt and cover with water. Bring to a boil, and then reduce to a simmer to cook through and flavor the stock, 30 minutes. Transfer the chicken to a shallow dish until cool enough to handle. Strain the stock and reserve the meat.
Place the dried mushrooms in a small pot and cover with 1 to 1 1/2 cups of the stock. Bring to a simmer until the mushrooms are reconstituted, 10 to 15 minutes.
Remove the meat from the chicken bones, discarding the skin and bones, and pull into small pieces. You will need only the dark meat. (Reserve the white meat for another use, such as Creamy Chicken and Noodles.)
Heat the EVOO over medium-high heat in the pot used to cook the chicken. Cook the pancetta to render, 2 to 3 minutes. Stir in the chopped onion, garlic and rosemary, sprinkle with salt and pepper and cook until the onions are tender, 7 to 8 minutes. Stir in the tomato paste, pour in the wine and cook until reduced by half. Return the pulled chicken into the pot with 2 cups of the reserved stock. (Reserve the remaining chicken stock for another recipe, such as Creamy Chicken and Noodles.)
Chop the reconstituted mushrooms and add them into the sauce along with their liquid. Simmer to thicken the sauce, 20 to 30 minutes.
Bring a large pot of salted water to boil and cook the pasta to al dente. Reserve 1 cup of the starchy cooking liquid just before draining. Toss the pasta with the sauce and starchy water. Mix in some pecorino, and serve with more to pass at the table. Garnish with grassy flat-leaf parsley.
Cook's Note: Cool and store the sauce for a make-ahead meal. Bring to room temperature before reheating over medium heat. Cook the pasta right before serving.