Bring a large pot of water to boil. Add the chicken breasts, cover, and cook for 15-20 minutes. When the chicken is done, remove from heat, drain water, let cool, and shred with two forks.
Chop the cabbage into very thin pieces, like it would be for coleslaw. I use a mandolin. Peel and grate the carrots. Cut off the skin, remove the seeds, and grate the papaya. Roughly chop the cilantro and green onions. Toss the chicken and vegetables in a large bowl and keep chilled.
Mince the garlic and chili peppers. Place garlic and peppers in a small mixing bowl with the soy sauce, vinegar, sugar, lime juice, oil, and fish sauce. Whisk until smooth. Add the peanut butter and water and whisk again until smooth and creamy.
Toss the salad with the dressing. Add the crushed peanuts. Serve chilled. Leftovers can be stored in the fridge for up to one day. For best results, keep the leftover salad and dressing separate until ready to serve.