Melt butter in a large saucepan over medium heat. Add onion and sweat until soft but not browned. Add potato and stock and bring to a gentle simmer. Partially cover pot and cook until potatoes are soft, 15–20 minutes. When potatoes are almost cooked, add watercress. Cook for no more than 5 minutes.
Carefully transfer to a blender (or use an immersion blender) and liquidize soup. Return to pot and add half and half. Season with salt and pepper and serve hot with crusty bread.
NOTE: St. Brendan allegedly lived to the age of 180 by eating just watercress.