If you're making your own peanut sauce, follow the recipe directions and then refrigerate for later. This can be made 3 to 4 days ahead.
In the bowl of your slow cooker, whisk to combine lime juice, soy sauce, brown sugar, oil, fish sauce, chili paste, garlic, ginger, and white parts of the onions.
Add in the chicken and stir to coat with sauce. Cook, on high for 4 hours or 8 on low.
Once the chicken is cooked, remove the chicken from the slow cooker and cut it into smaller pieces or shred it between two forks.
In a large bowl, combine the cooked chicken, shredded cabbage, carrots, bell peppers, peanuts, green parts of the onion, and cilantro.
Serve by scooping chicken and veggie mixture inside lettuce leaves and drizzle with peanut sauce.