Finely chop onion. Grate carrot. Set carrot aside.
Heat olive oil in large stock pot. Add chopped onion and saute for 3-5 minutes.
Peel and chop potatoes.
Add water, broth, salt, pepper, cayenne pepper, and chopped potatoes to the stock pot. Bring to a boil and simmer over medium heat for 30 minutes until the potatoes are tender.
When potatoes are tender, pull pan from heat and allow the soup to stop boiling and slightly cool. Insert the chopping end of an immersion blender completely under the liquid. Gently pulse to blend some of the potatoes. Move the blender around until you have blended the potatoes in 3-4 spots. Leave about half of the potatoe chunks intact. Remember the soup is hot so do this carefully.
Add chicken, milk and grated carrot to soup. Return soup to a boil over medium heat.