Cook the rice as the label directs. Set aside off the heat, 5 minutes, then fluff with a fork.
Meanwhile, heat the vegetable oil in a dutch oven or other large pot over medium-high heat. Season the chicken thighs generously with salt and pepper. Working in batches if necessary, add the chicken to the pot, skin-side down, and cook until lightly browned and the fat has rendered, 3 to 4 minutes. Flip and cook until browned on the other side, 3 to 4 minutes. Remove the chicken to a plate.
Reduce the heat under the pot to medium. Add the onion, garlic and ginger and cook until softened, about 2 minutes. Add the fish sauce and cook until reduced by half, about 3 minutes. Add the chicken broth and bring to a boil. Add the chicken and chayotes to the pot and reduce the heat to medium to maintain a steady simmer. Cover and cook until the chicken is cooked through, about 15 minutes. Add the bok choy, cover and cook until tender, about 5 more minutes.
Divide the stew among bowls. Serve with the rice.
This Filipino dish traditionally contains green papaya, but many versions, like this one, call for chayote instead.