1 Put the chicken in a large (at least 8 quart) stock pot and add 8 cups of water. Bring to a boil over high heat and then reduce the heat to a vigorous simmer. Simmer for 10 minutes, frequently skimming off the foam that floats to the surface.
2 Add all the vegetables, the garlic, the cilantro, and the bouillion cubes to the pot, along with the salt and pepper. Stir a few times to distribute the vegetables and submerge as many of the solids as possible. When the broth returns to a boil, lower the heat, partially cover the pot, and simmer, stirring once or twice for 1 1/2 hours. Taste for salt and add more if needed.
3 Using tongs or a slotted spoon, pick out the chicken pieces and put them on a large plate. Stir the soup with a large spoon, breaking up some of the potatoes to thicken the soup slightly.
4 When the chicken is cool enough to handle, pull the meat off the bones and shred it by hand. Discard the bones and tendons. When the soup is ready to serve, return the chicken to the soup.
5 Pulse all the aji ingredients in a food processor until they're finely minced. Transfer to a serving bowl.
6 Serve the soup with the aji, sliced avocados, sour cream, and cilantro in separate bowls. Let people add garnishes to their own servings.