In a large pot or Dutch oven, heat a little bit of oil (vegetable or canola) over medium heat. Add the chopped onion and cook until it's softened, about 5 minutes. Then, add the minced garlic and sliced ginger, and cook for another minute until fragrant.
Pour in the chicken broth. Add the sliced carrots, celery, and mushrooms. Bring the mixture to a boil, then reduce the heat to low, cover, and simmer for at least 15 minutes.
Stir in the cooked, shredded chicken (or tofu). Add the soy sauce, sesame oil, and white pepper. Taste and add salt as needed.
While the soup is simmering, cook the noodles according to the package directions. Drain the noodles and rinse them with cold water to prevent sticking.
In the last few minutes of simmering, add the chopped bok choy to the soup.
Divide the cooked noodles among bowls. Ladle the hot soup over the noodles.
Top with your favorite garnishes – sliced green onions, a drizzle of chili oil, toasted sesame seeds, or a perfectly soft-boiled egg.