Heat oil in a large saucepan over medium heat. Add onions, celery, and bell pepper, and cook until soft, 5 to 7 minutes. Add 3 cups of broth and tomatoes. Bring to a simmer, and then reduce the heat to medium-low and simmer gently until the flavors blend, about 5 minutes.
Puree with a stick blender or in an upright blender or food processor (in batches if necessary). Return soup to pan and stir in peanut butter, salt, cayenne, and chicken. Heat through, 2 to 3 minutes.
Tip: To make this soup even more quickly, buy a rotisserie chicken and cut the meat into cubes.