Place the peanut butter in a microwavable bowl, and heat for 15 seconds. Stir, and return to the microwave for another 10 seconds, and stir again.
Add the soy sauce, brown sugar, ginger and sesame oil to the bowl. Stir until smooth and completely combined.
Add the chicken breasts to the bowl, and turn to coat in the peanut sauce.
Place the chicken in the base of a slow cooker. Spread any peanut sauce remaining in the bowl over the chicken.
Place the potatoes beside the chicken, and top with the carrots and green beans.
Cook on the lowest setting for 4-5 hours.
Remove the carrots, green beans and potatoes from the slow cooker and arrange on plates.
Shred the chicken between two forks, tossing it in the peanut sauce, and serve. Garnish with cilantro leaves and additional peanuts.
Spoon any additional peanut sauce over the veggies.