Place the chicken parts in a large pot and add ½ tablespoon of salt. Cover with cold water. Bring to a boil, then reduce to a simmer, skimming off any impurities. 

Add the carrots and onion, and simmer for 10 more minutes, skimming as needed. Add 1 of the potatoes and cook for an additional 20 minutes. Remove the chicken and reserve the soup.

While the chicken and vegetables cook, heat ½ cup of the oil until it shimmers in a large pot or Dutch oven. Working in 2 batches, fry the remaining potatoes until browned on all sides, 15 minutes per batch. Repeat with the remaining ⅓ cup of oil and potatoes. Transfer the potatoes and oil to a bowl, and wipe out the pot.

In the same pot, arrange the chicken pieces in a single layer, skin side down. Arrange the potatoes on top, pouring ⅓ cup of the potato cooking oil over top. Sprinkle with the remaining ½ tablespoon of salt and the cinnamon. Add 2 ½ cups of the reserved soup, just enough to come halfway up the chicken. 

Bring the soup to a boil, then lower the heat to a gentle simmer and continue cooking uncovered for 45-50 minutes or until the liquid has reduced to a thick sauce and the chicken is cooked through, basting occasionally. Serve hot.