Place the chicken in a large pot and cover with 12 to 16 cups of water. Ensure the chicken is submerged. Add the garlic, jalapeño, quartered onion, and 2 bay leaves. Add 4 teaspoons kosher salt. Bring to a boil. Reduce the heat to low and cover. Simmer for 1 hour and 30 minutes or until the chicken is completely cooked through.
Remove the chicken from the pot and transfer it to a large bowl. Turn the heat on the broth to medium and simmer rapidly for 20 to 30 minutes to reduce the broth. You should be left with 8 to 10 cups of broth.
Strain the broth through a fine-mesh sieve into a large bowl. Remove the head of garlic and set it aside. Discard the rest of the solids. 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 into bite-sized pieces. Reserve skin and carcass for stock, if desired.
Carefully pop the cloves of garlic out of their paper and mash them into a paste. Set aside.