Place a single layer of the chicken pieces or scraps, meat side down, in a 6- to 8-quart pan over medium-high heat. Cook without disturbing until meat is well browned, 7 to 10 minutes (take care not to scorch). Using tongs or a long-handled fork (meat may stick; pry it up from pan bottom), turn pieces and cook until second side is browned, about 5 minutes longer. Reduce heat to medium.Pour 2 cups water into pan and stir, scraping pan bottom to loosen browned bits.
Add remaining 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 stock 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 stock through strainer. Allow liquid to drip from solids, but do not press to extract more; discard solids. Stir stock occasionally until cool, 10 to 20 minutes; cover and chill until stock 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 stock and discard. Cover stock and chill up to 4 days, or pour into jars or ice cube trays and freeze up to 3 months.