Heat the olive oil in a large pot over medium heat. Add the onion, garlic, and ginger, and cook for 2 minutes until the onion softens and the ginger smells fragrant.
Add the carrots and stir for 1 minute. Pour in the chicken broth and bring the mixture to a gentle boil.
Place the chicken breasts into the pot. Reduce the heat to a simmer and cook for 15 minutes until the chicken is cooked through.
Remove the chicken from the pot, shred it with two forks, and return it to the broth.
Stir in the fish sauce and lime juice. Taste and add black pepper if needed.
Ladle the soup into bowls and top each serving with cilantro and green onions.