In a large pot, heat the vegetable oil over medium heat. Add the diced onion and sauté for about 3-4 minutes until softened.
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 pieces to the pot and reduce the heat to a simmer. Cook for about 10 minutes, or until the chicken is cooked through.
Add the chopped bok choy, sliced carrots, and mushrooms to the pot. Simmer for another 5-7 minutes until the vegetables are tender.
Stir in the soy sauce and sesame oil. Season with salt and pepper to taste.
Serve hot, garnished with sliced green onions.