In a 3-quart saucepan, combine water, potatoes, carrots, and onion. Bring to boiling; reduce heat. Cover and cook for about 15 minutes or just until vegetables are tender. Don't drain.
Meanwhile, in a screw-top jar, combine milk and flour; cover and shake until well combined.
Stir milk mixture into vegetable mixture. Cook and stir until mixture is thickened and bubbly.
Add cheddar cheese, ham, and American cheese. Cook and stir over low heat until cheeses are melted. Season with pepper. Makes 6 side-dish servings.
Double the recipe, if you like.