Generously season each piece of chicken with salt and pepper. Heat the olive oil in a large pot over high heat until it begins to smoke. Carefully add chicken pieces in a single layer, skin side down, to brown each piece. After about 8 minutes, turn each piece and brown them on the second side. When the chicken pieces are a deep brown on both sides, place the onions, celery, and carrots in the pot and cook until deeply browned, about 15 minutes, stirring only occasionally.
Add the mushrooms, if using. Add the wine and bay leaves. With a wooden spoon, scrape the bottom of the pot to remove any caramelized bits. Cook over high heat until about 1/4 cup of wine remains, about 5 minutes. Add the water. When the water comes up to a simmer, skim off any foam or impurities that rise to the top. Lower heat to maintain a simmer, but do not let the broth boil.
Simmer the broth uncovered for 2 1/2 to 3 hours, occasionally skimming any foam and impurities that rise to the top. The chicken meat will be tender and falling off of the bone. Remove the chicken pieces from the broth and reserve. Pour the broth through a colander lined with a tea towel (or a triple thickness cheese cloth) and set over a large, clean pot (or strain the broth into a large bowl and return the broth to the pot). Discard the vegetables. You should have about 8 cups of broth. Keep the broth warm. Taste and season really well (this is a simple dish that requires plenty of salt and pepper).
When the chicken is cool enough to handle, shred the chicken meat from the bones. Discard the bones. Place the shredded chicken in a large bowl.
Cook the pasta in a large pot of boiling salted water for about 5 minutes, until just shy of al dente, and then drain it through a colander. Pour the warm pasta directly into the bowl with the shredded chicken. Add the parsley and season with salt and pepper. Drizzle with olive oil and stir. Add 1 cup of reserved broth to moisten the mixture.
Mix well and divide evenly among 8 serving bowls. Ladle 1 cup of warm broth into the bowl. Scatter some Parmesan over each bowl of soup. Pass chili flakes for extra flavor. Serve warm with both a spoon and a fork.