Fill a small saucepan half full with water, add 1 tsp salt and bring to boil.
Add the chicken breasts. Bring the water back to boil, remove the pan from the heat and cover tightly. Let stand for 20 minutes.
Remove the chicken from the pan and discard the water. The chicken should be firm yet still yield a bit to the touch. When cool enough to handle, shred with your fingers into thin pieces, pulling the meat along its natural grain.
Put the chicken in a large bowl and let it cool to room temperature.
Meanwhile, put the onion in a small bowl and add the vinegar just to cover. Set aside for 15 minutes.
Drain well and add it to the bowl with the chicken, along with the cabbage, carrot and cilantro.
To make the dressing, put the chili, garlic, sugar and salt together and mash using a pestle and mortar, or a small blender. Scrape the paste into a bowl and add the fish sauce and rice vinegar. Keep stirring until the sugar and salt have dissolved.
Just before serving, pour the dressing over the salad and toss. Serve.
Notes
You can make the dressing ahead of time and keep it in the fridge for about 2 weeks.