Heat the vegetable oil in a large saucepan oil over medium-high. Add the ginger, garlic, scallions and celery and cook, stirring, until softened, about 3 minutes. Add the chicken broth and 2 cups water, bring to a simmer and cook until the vegetables are tender, about 5 minutes.
Add the frozen dumplings or wontons, bok choy, chicken and soy sauce and simmer until warmed through, 3 to 5 minutes; season with salt. Divide the soup among 4 bowls, then drizzle with sesame oil and top with scallions.