Place the chicken broth, onion halves, ginger slices, star anise, and cinnamon stick in a large pot. Bring to a simmer over medium heat and cook for 20 minutes to build flavor.
Add the chicken breasts to the simmering broth. Poach for 15 minutes until the chicken reaches 165°F and is cooked through. Remove the chicken and set it aside to cool slightly.
While the chicken cooks, prepare the rice noodles according to package directions in a separate pot. Drain and divide them among four bowls.
Shred the cooled chicken with two forks. Return the shredded chicken to the pot and stir in the fish sauce.
Ladle the hot broth and chicken over the noodles in each bowl. Top with bean sprouts, cilantro leaves, Thai basil, and sliced green onions. Serve with lime wedges on the side.