Whisk hoisin, soy sauce, rice vinegar, orange juice, ginger and garlic in a 4-quart slow cooker. Add whole chicken breasts and turn them through the sauce.
Cover and cook on low for 3 to 4 hours, checking from 3 hours and removing chicken as it reaches 165°F.
Pull chicken into long coarse shreds. Pour cooking liquid into a saucepan and simmer for 6 to 8 minutes until reduced, then stir in sesame oil.
Toss chicken with only enough reduced sauce to moisten it without leaving liquid pooled in the bowl. Keep any extra sauce separate for another use.
Fill the 18 dry lettuce leaves with chicken and divide carrot, cucumber and peanuts evenly over them. Serve 3 wraps per person while the chicken is warm and the leaves are crisp.