Line a plate with a paper towel or a cotton kitchen towel. Set a medium-ish skillet, preferably cast-iron, over medium heat. Spoon enough lard or coconut oil into the skillet so that it reaches about 1⁄2 inch (1 cm) up the side of the skillet when it melts.
Once the fat has melted completely and begins to shimmer, test the oil by dropping a tortilla strip into the hot fat. The oil is ready when the tortilla immediately sizzles and then crisps and turns a golden brown in about 60 to 90 seconds. Working in batches so as to take care not to crowd the pan, fry the tortilla strips, turning as necessary, until crisp and golden brown. Using a slotted spoon, transfer the tortilla strips to the towel-lined plate and let them cool. Turn off the heat.
Nestle the whole chicken in a large stock pot and pour enough water into the pot to cover the chicken by 2 inches (5 cm). Bring the water to a gentle boil over medium-high heat and then immediately reduce the heat to medium-low, partially cover, and very gently simmer until the chicken is cooked through and the meat shreds easily with a fork, about 2 hours.
When the chicken is cooked through, turn off the heat. Remove the chicken from the pot and plonk it on a plate to rest until it’s cool enough to handle. Remove and discard the skin. Pull the meat from the bone and shred it with a fork.
Strain the cooking liquid, or stock, through a fine-mesh sieve, discarding the solids. Wipe out the pot and then return the strained stock and reserved chicken meat to the pot. Stir in the onion, rice, and 2 teaspoons salt and then bring to a simmer over medium heat. While the soup warms, squeeze the juice of 1 to 2 limes into the pot. Continue cooking until the onion is soft and translucent and the chicken is warmed through, about 10 minutes. Taste the finished soup and if it tastes bland add a little more lime juice and salt as needed.
While the soup is cooking, finely chop the remaining 2 limes, peel and all. [Editor’s Note: If the only limes you can find have thick peels, simply juice the remaining 2 limes rather than chop them.] Ladle the soup into bowls, being certain to scoop some rice from the bottom of the pot, and serve with the lime, sliced jalapeño, crumbled Cotija cheese, sliced avocado, and tortilla strips on the side for each person to add to his or her bowl as desired.