Place the chicken in a large stockpot or Dutch oven and cover with the chicken stock and 2 quarts water. Add the onion quarters, roughly chopped carrots and celery, thyme, parsley stems, and the bay leaf. Bring mixture to a boil, partially cover and reduce heat to a simmer. Simmer for 1 hour. The chicken should easily pull away from the bones.
Using tongs, remove the chicken from the broth and set aside until cool enough to handle. Strain the cooking liquid through a fine-meshed sieve, lined with cheesecloth, discarding the vegetables and reserving the stock. Pull the chicken meat off the bones, discarding the bones, skin, and any fat. Shred and reserve meat.
Melt the butter in a large stockpot over medium-high heat. Add mushrooms and cook until browned, about 3 minutes. Add the diced onions, carrots, and celery and cook until the vegetables are softened, about 2 minutes. Stir in reserved cooked chicken. Add reserved chicken stock and bring to a boil over high heat. Season the soup with salt and pepper. Add vermicelli and simmer until noodles are cooked through, 10 to 12 minutes. Remove from the heat and stir in parsley. Serve hot.