This is the base for the ultimate comfort food in wintertime... or any time you might be feeling sick.
In a 10- or 12-quart stockpot, heat oil over high. Add chicken parts and cook, turning frequently, until browned on all sides, 15 minutes.
Pour out fat. Add whole chicken, carrots, celery, onion, peppercorns, parsley, and enough cold water to cover ingredients by 1/2 inch (about 12 cups).
Bring to a boil over high. Reduce heat and simmer until chicken is cooked through, 40 minutes, skimming fat and foam and flipping chicken once.
Transfer whole chicken to rimmed baking sheet and let cool, about 10 minutes.
Pour liquid through a paper-towel-lined sieve into a large bowl (you should have 10 cups broth). Discard vegetables and chicken parts.
Shred chicken into bite-size pieces, discarding skin and bones (you should have 4 cups meat).
Refrigerate broth and shredded chicken in separate airtight containers, up to 3 days. If desired, freeze broth up to 3 months.