How to Make It
Put the mulato chiles in a medium bowl and cover with the hot water; cover them with an inverted small plate to keep them submerged. Let the chiles soak until softened, about 30 minutes. Drain the chiles, reserving 1/2 cup of the soaking liquid. Stem, seed and coarsely chop the chiles.
Heat a large nonstick skillet. Season the chicken with salt and pepper and add the pieces to the skillet, skin side down. Cover partially and cook over moderately high heat until well browned on both sides, about 15 minutes. Transfer the chicken to a plate. Pour off all but 1 tablespoon of fat from the skillet and reserve for cooking the rice (optional).
Add the scallions, garlic, onion and chorizo to the remaining fat in the skillet and cook, stirring, until softened, about 4 minutes. Add the chopped chiles, wine, tomato, oregano and the reserved 1/2 cup of chile soaking liquid and simmer for 1 minute. Arrange the chicken in the skillet, skin side up. Tuck the chayote in between the pieces of chicken. Cover and cook over low heat until the chayote is tender and the chicken is cooked through, about 30 minutes.
Transfer the chicken and chayote to plates. Boil the sauce over high heat until reduced, about 4 minutes. Season with salt and pepper. Spoon the sauce around the chicken and serve with Achiote Rice.