How to Make It
In a small skillet, toast the coriander, allspice, cumin and peppercorns over moderately high heat, shaking the pan, until fragrant, about 40 seconds. Transfer to a plate to cool, then finely grind in a spice grinder or mortar. Stir in the turmeric.
In a large bowl, toss the chicken with 1 1/2 tablespoons of the lime juice and season with salt and pepper. In a large enameled cast-iron casserole, melt the butter in the oil. Add the chicken and cook over moderately high heat without turning, until browned, about 3 minutes. Turn the chicken and add the onion, scallions and shallot. Reduce the heat to moderate and cook until the chicken is browned, about 4 minutes. Add the ground spices and the garlic, parsley and thyme and cook, stirring, until fragrant, about 2 minutes.
Add the remaining 1 tablespoon lime juice and 2 cups of water to just cover the chicken. Simmer over very low heat for 30 minutes. Add the tomatoes, chayote, eggplant and chile and season with salt and pepper. Cover and simmer until the chicken and vegetables are tender, about 30 minutes. Transfer the chicken to a plate. Boil the stew over moderately high heat until the liquid is slightly reduced, about 4 minutes. Discard the thyme. Return the chicken to the casserole, season with salt and pepper and serve with rice.