Place the chicken, jalapeños, and halved onion in a large pot and cover with water. Add a generous sprinkle of salt. Bring to a boil. Reduce heat, cover, and simmer for 1 hour and 30 minutes.
Using tongs, carefully remove the bird and transfer it to a large bowl. Let cool.
Turn the heat on the broth in the pot to a rapid simmer. Let simmer for 30 minutes or until it reduces to about 8 cups of broth. It will darken in color as it cooks down. Adjust the heat as needed if it reduces too quickly.
Strain the broth into a large bowl through a fine-mesh sieve. Discard solids.
Carefully 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 the meat into bite-sized pieces. Reserve or freeze the carcass for stock later if you like.