In a 6- to 8-quart pan over medium-high heat, combine chicken, onion, carrot, celery, parsley, peppercorns (if using), bay leaf, and thyme. Add cold water just to cover. Bring to a simmer, then lower heat to maintain simmer and cook, occasionally skimming and discarding foam from surface, until liquid is golden and has a deep chicken flavor, about 2 hours. Do not allow broth to boil; the surface should barely be disturbed by small bubbles.
If a layer of fat forms on the surface, skim off and discard. If liquid drops below level of chicken and vegetables, add more cold water just to cover.
Place a fine strainer over a large bowl nested in ice water in the sink. Ladle or carefully pour broth through strainer. Allow liquid to drip from solids, but do not press to extract more; discard solids. Stir broth occasionally until cool, 10 to 20 minutes; cover and chill until broth is cold and any fat on surface is firm, at least 6 hours, or up to 1 day.
With a spoon, skim fat from surface of broth and discard. Cover broth and chill up to 4 days, or pour into jars or ice cube trays and freeze up to 3 months.