Heat up extra-virgin olive oil in a deep saucepan. Add the prosciutto or salami and cook until crisp. To make the 'soffritto', add the onion, celery, carrot, chile pepper, salt, and black pepper. Saute until the onion becomes soft, and then add the potatoes. Stir to prevent the potatoes from sticking to the pan. Add the water and tomato puree.
Add the spaghetti to the pot. Mix together well and add the Pecorino cheese. Add water, a little at a time periodically and cook for a few minutes until the soup is creamy and thickened.
Drizzle with olive oil and sprinkle with Parmigiano cheese. Allow the soup to cool and thicken. Serve the soup at room temperature.