Heat 1 tablespoon neutral oil over high heat in a large, heavy-bottomed pot. Add the onion and jalapeños and cook for 2–3 minutes until they begin to char. Add the garlic, cut side down, for 1 minute until it begins to brown.
Place the chicken on top and pour over 8–10 cups of water, enough to completely submerge the chicken. Season with a teaspoon or two of salt. It will cook down and become quite salty, so there is no need to go too heavy with it.
Bring to a boil, reduce heat, cover, and simmer for 1 hour. Check the broth occasionally and skim off any foam or scum that floats to the top.
After 1 hour, remove the chicken from the pot and transfer it to a bowl.