Place the chicken and vegetables in a large stockpot over medium heat. Pour in only enough cold water to cover (about 3 quarts); too much will make the broth taste weak. Toss in the thyme, bay leaves, and peppercorns, and allow it to slowly come to a boil. Lower the heat to medium-low and gently simmer for 1 to 1½ hours, partially covered, until the chicken is done. As it cooks, skim any impurities that rise to the surface; add a little more water if necessary to keep the chicken covered while simmering.
Carefully remove the chicken to a cutting board. When its cool enough to handle, discard the skin and bones; hand-shred the meat into a storage container.
Carefully strain the stock through a fine sieve into another pot to remove the vegetable solids. Use the stock immediately or if you plan on storing it, place the pot in a sink full of ice water and stir to cool down the stock. Cover and refrigerate for up to one week or freeze.
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