Place the chicken in a large pot with the thyme sprigs, bay leaves, kosher salt, garlic, and onions. Cover with 14 to 16 cups of water (enough to immerse the chicken). Bring to a boil. Reduce the heat to low and cover. Simmer for 1 hour 30 minutes or until the chicken is completely cooked through.
Using tongs, carefully transfer the chicken to a large bowl. Return the broth to a low boil over medium heat. Cook, uncovered, for 20 to 30 minutes to reduce the broth to about 12 to 13 cups.
Strain the broth through a sieve into a large bowl. Discard the spent aromatics. Wipe out the pot and return it to the stove.
Once the chicken is cool enough to handle, pick the meat from the bones and shred it with your fingers or dice it with a knife. Reserve the skin and carcass for stock if you like.