Prepare the Aromatics: Heat the vegetable oil in a large pot or Dutch oven over medium heat. Add the grated ginger and minced garlic, sautéing for 1-2 minutes until fragrant but not browned.
Simmer the Base: Pour in the chicken broth and soy sauce. Add the whole chicken breasts (or thighs) to the pot. Bring the mixture to a boil, then reduce the heat to low. Cover and simmer for about 15-20 minutes, or until the chicken is fully cooked through.
Cook the Rice: Remove the cooked chicken from the pot and set it aside to cool slightly. Add the rinsed jasmine rice to the simmering broth. Stir well, cover, and cook for 15-20 minutes, or until the rice is tender and has absorbed some of the liquid.
Shred and Combine: While the rice cooks, shred the chicken using two forks. Once the rice is done, return the shredded chicken to the pot.
Add Greens: Stir in the chopped bok choy. Let it cook for 2-3 minutes until the greens are wilted but the stems remain crisp-tender.
Serve: Remove the soup from the heat. Stir in the sesame oil. Ladle into bowls and garnish generously with sliced green onions. Season with salt and pepper to taste.