Rinse the chicken well under cool running water and remove the neck, gizzard, heart, and liver from the cavity if present. Place the chicken in a large stockpot or Dutch oven along with any parts (except the liver; it can give the stock a bitter taste), and cover with the stock and water. Add the remaining ingredients. Bring to a boil, partially cover, and reduce the heat to a simmer. Cook for 1 hour; at that point, the chicken should easily pull away from the bones.

Using tongs, remove the chicken from the stock and set aside until cool enough to handle. Strain the cooking liquid through a fine-mesh strainer lined with cheesecloth, discarding the vegetables. Pull the chicken meat off the bones, discarding the bones, skin, and any fat. Shred and reserve the meat, and refrigerate until needed. Cool the stock, then store in an airtight container in the refrigerator for up to 4 days or the freezer for 6 months.