Place the chicken breasts in a large saucepan. Add a pinch of salt. Cover with cold water by at least 3 cm. Bring to a gentle boil over medium high heat. Reduce to low and simmer for 10 minutes.
Turn off the heat. Cover with a lid and leave the chicken in the hot liquid for 10 minutes. Remove and transfer to a plate. Cool for about 20 minutes, or until easy to handle.
Shred the chicken into rough bite size pieces. Cover and chill if needed. You can prepare the chicken up to 2 days ahead.
Add cabbage, carrots, spring onions, herbs and chilli to a large bowl. Add half the chicken, half the peanuts, half the fried shallots and some of the Nuoc Cham Dressing. Toss well.
Transfer to a serving plate. Top with remaining chicken, peanuts and fried shallots. Drizzle with more dressing. Serve straight away.