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