Bring broth and water to a simmer over medium-high heat in a large soup kettle. Meanwhile, separate chicken meat from skin and bones; reserve meat. Add skin and bones to the simmering broth. Reduce heat to low, partially cover and simmer until bones release their flavor, 20 to 30 minutes.
Strain broth through a colander into a large container; reserve broth and discard skin and bones. Return kettle to burner set on medium-high.
Add oil, then onions, carrots and celery. Saute until soft, about 8 to 10 minutes. Add chicken, broth and thyme. Bring to a simmer. (Can be refrigerated up to 3 days in advance. Return to a simmer before adding the extras of your choice.)