Heat oil in Dutch oven over medium heat. Add carrot, celery, and onion; cook 5 min or til tender, stirring frequently. Add ginger, 1/4 t salt and garlic; cook 1 min, stirring constantly. Add 1/4 t salt, water, soy sauce, chile sauce, broth, and chicken. Bring to a boil; reduce heat and simmer 1 hour. Remove chicken from pan and cool slightly.
Remove chicken from bones. Shred meat with 2 forks. Add chicken to pan, bring to a boil. Add soba, cook 5 minute Stir in bok choy, reduce heat and simmer 3 minute Stir in 1/4 t salt, green onions, basil and pepper; cook 2 minute Remove from heat and stir in lime juice.