Heat oil in a large pot over medium heat. Add onion, ginger and garlic; cook, stirring, until the onion starts to turn translucent, about 3 minutes. Add chicken and broth; cook, stirring, until the chicken is just cooked through, about 5 minutes. Add papaya (or chayote), malunggay (or bok choy), fish sauce, salt and pepper; continue simmering until the vegetables are tender and the flavors have melded, about 5 minutes more.