Place chicken and broth in a large pot. Bring to a boil over high heat. Reduce heat to medium-low. Simmer, uncovered, for 30 minutes. In a large pan, cook chilies over low heat, turning frequently, until they change color and become fragrant. Stir in tomatillos, onion, garlic, cumin and 1 cup water. Bring to a boil over high heat. Cook, uncovered, for 5 minutes. Remove pan from heat. Let cool. Place chili-tomatillo mixture (mole) in a blender container. Puree until smooth. Return pureed mixture to pan. Cook over medium heat, stirring occasionally, until thickened, about 5 minutes. Stir the thickened mole into chicken and broth. Simmer, uncovered, for 15 minutes. Meanwhile, in a medium saucepan, bring 2 cups water to a boil over high heat. Add chayotes and green beans. Cook 5 minutes. Drain. Serve with chicken and mole.