Combine stock, chicken, ginger, and garlic in a large pot or Dutch oven. Bring to a boil. Reduce heat to low and simmer, covered, until chicken is cooked through, 25 to 30 minutes. Add mushrooms, soy sauce, and sesame oil and simmer, covered, until mushrooms are tender, 8 to 10 minutes. Remove chicken and shred with two forks.

Add noodles to pot and cook, stirring occasionally, until beginning to soften, 2 to 4 minutes. Stir in chicken, bok choy, and snow peas. Cook until vegetables are crisp-tender, 1 to 2 minutes. Stir in scallions just before serving.