Place chicken in a large stockpot or Dutch oven and cover with stock and water. Add quartered onion, chopped carrots, chopped celery, thyme, parsley, and bay leaf. Season with salt and pepper. Bring mixture to a boil, partially cover, and reduce heat to a simmer. Cook for 1 hour; chicken should easily pull away from the bones.

Using tongs, remove chicken from broth and set aside until cool enough to handle. Strain cooking liquid through a fine-meshed sieve lined with cheesecloth, discarding vegetables and reserving cooking liquid. Pull chicken meat off bones, discarding bones, skin, and any fat. Shred and reserve meat.

Add oil to a large stockpot over medium-high heat. Add mushrooms and cook until browned, 3 to 4 minutes. Add diced onions, sliced carrots, and diced celery and cook until vegetables are softened, 2 to 4 minutes. Stir in butter and reserved cooked chicken. Add reserved cooking liquid and bring to a boil over high heat, then reduce to a simmer and cook for 30 minutes. Add 2 1/4 teaspoons salt, 1/2 teaspoon pepper, and vermicelli to the pot and simmer until noodles are cooked through, 10 to 12 minutes. Remove from heat and stir in parsley leaves. Serve hot.