Heat the oil in a heavy large saucepan or Dutch oven over medium heat. Add the celery, carrots and onion. Saute, stirring, until the vegetables are wilted, about 5 minutes. Reduce the heat to very low; sprinkle the vegetables lightly with salt; cover the pan and sweat the vegetables, stirring occasionally, until tender but not browned, about 15 minutes.

Add the broth and potatoes along with a small rosemary sprig - there should be enough broth to cover all vegetables. If not, add additional broth or water. Heat to simmering; cover the pan and simmer until the potatoes are very tender, about 30 minutes.

Discard the rosemary sprig. Puree the soup, either in a blender, with a hand-held immersion blender or using a food mill fitted with the fine disk. Use care as the soup will be hot. Return to the pan, season to taste with salt and pepper and heat to simmering. Ladle into warmed bowls. Garnish with a small rosemary sprig and pass Agliata separately, if desired