In a large pot, heat the olive oil over medium heat. Add the diced onion and sauté until translucent, about 5 minutes.
Stir in the minced garlic and cook for an additional minute until fragrant.
Add the chicken pieces to the pot and cook until they are no longer pink, about 5-7 minutes.
Pour in the chicken broth and water, then add the grated ginger, soy sauce, and sesame oil. Bring the mixture to a boil.
Once boiling, reduce the heat to a simmer and add the chopped bok choy, sliced carrots, and mushrooms. Cook for 10-15 minutes until the vegetables are tender.
Season with salt and pepper to taste.
Serve hot, garnished with chopped green onions.