Combine first 4 ingredients in a medium saucepan; bring to a boil. Cover, remove from heat, and let stand 15 minutes or until chicken is done. Remove chicken from pan, and discard peppercorns, bay leaf, and cooking liquid. Shred chicken; place in a large bowl.
Combine vinegar and next 5 ingredients (vinegar through salt), stirring with a whisk. Pour over chicken; let stand 5 minutes. Add soba noodles and the next 4 ingredients (noodles through chopped basil) to chicken mixture, and toss well. Sprinkle with peanuts. Garnish with lime wedges, if desired.