Using a heavy cleaver, chop each chicken leg into four or five pieces. (You can also have your butcher do this for you or you can just cut the chicken into thigh and drumstick portions.) Set the chicken aside.
In a small stockpot or large saucepan, heat the oil over medium heat. Stir in the onion, red pepper, garlic and achiote (if using). Cook, stirring occasionally, until the onion and pepper are very soft, about 15 minutes, reducing the heat as necessary to prevent the onions from browning.
Stir in the turmeric, thyme, espelette, salt and pepper and cook, stirring, for one minute. Add the tomato paste and cook, stirring, for two minutes. Stir in the broth and quinoa. Add the chicken pieces and chayote. Bring to a boil, reduce the heat, partially cover the pot, and simmer until the chicken is cooked through and no longer pink inside and the quinoa has burst open, 40 to 45 minutes. Serve in deep bowls with lots of crusty bread to sop up the delicious juices.