1. Marinate chicken in 1/4 cup teriyaki sauce for 20 minutes.
2. Preheat an oiled grill or grill pan. Grill chicken over medium heat, turning once, for 15 minutes or until cooked through. Transfer to cutting board, and tent with foil; let stand.
3. Combine broth, water, ginger, garlic, and remaining 1 1/2 tablespoon teriyaki sauce in a saucepan; bring to a boil. Add mushrooms, carrots, and scallions; simmer for 10 minutes.
4. Boil soba for 4 minutes or until just tender. Drain and divide among serving bowls. Stir sesame oil, basil, and cilantro into broth; divide among bowls. Slice chicken, and add to bowls along with radishes.