In a large pot, heat the olive oil over medium heat. Add the diced onion and sauté for about 3-4 minutes until translucent.
Stir in the minced garlic and grated ginger, cooking for an additional minute until fragrant.
Pour in the chicken broth and water, bringing the mixture to a boil.
Add the chicken breasts to the pot and reduce the heat to a simmer. Cook for about 15-20 minutes, or until the chicken is cooked through.
Remove the chicken from the pot and shred it using two forks. Return the shredded chicken to the pot.
Add the chopped bok choy, sliced carrots, and mushrooms to the soup. Stir in the soy sauce and sesame oil. Simmer for another 5-7 minutes until the vegetables are tender.
Season with salt and pepper to taste.
Serve hot, garnished with sliced green onions.