Add the chicken breasts, green beans, soy sauce, fish sauce, garlic, ginger, red pepper flakes, and chicken broth to the slow cooker. Stir to coat the chicken and beans evenly.
Cover and cook on low for 5 hours, until the chicken reaches an internal temperature of 165°F and shreds easily with a fork.
About 30 minutes before the chicken is done, bring the water to a boil in a medium pot. Stir in the jasmine rice, reduce heat to low, cover, and simmer for 15 minutes until the water is absorbed.
Remove the chicken from the slow cooker and shred it with two forks, then return it to the pot. Stir in the Thai basil leaves and lime juice. Let everything sit for 5 minutes so the basil wilts slightly.
Fluff the rice with a fork. Serve the chicken and green beans over the rice.