Toss the chicken in a 6- to 8-quart slow cooker with the sesame oil, grated ginger and a pinch each of salt and pepper. Stir in the rice to coat. Stir in the chicken broth, ginger slices and 4 cups water. Cover and cook on low until the chicken is very tender and the rice is thick, 7 to 8 hours. Stir gently a few times, then let stand, uncovered, 10 minutes. Discard the ginger slices.
Meanwhile, heat the vegetable oil in a large skillet over medium heat. Add the garlic and cook, stirring constantly, until golden and slightly crisp, 5 to 6 minutes. Remove with a slotted spoon to a small dish. Add the bok choy stems to the skillet, season with salt and cook, stirring, until crisp-tender, about 3 minutes. Add the bok choy leaves and cook until wilted, about 2 minutes.
Add the bok choy mixture and fish sauce to the slow cooker and stir gently, breaking the chicken apart slightly; season with salt and pepper. Divide among bowls. Top with the fried garlic, scallions, soy sauce and hot sauce. Serve with lime wedges.