In a medium pot, cover the chicken broth and bring to a boil. Add the chicken and potatoes; cover and simmer over medium-low heat until the potatoes are tender and the chicken is just cooked through, about 25 minutes.
Meanwhile, in a medium skillet, heat the olive oil over medium heat. Add the onion, chile and bell pepper and cook until softened, about 10 minutes.
Remove the chicken breasts from the broth and shred the meat; set aside. In a blender, puree half the broth and all the potatoes. Pour the mixture into the remaining broth and stir in the peanut butter until combined. Add the chicken and sauteed vegetables. Stir in a little water to thin the soup if desired. Heat gently. Season with salt, pepper and lime juice to taste.