Heat a deep, heavy bottomed pot over moderate heat. Add oil, garlic, onions, carrots, celery and bay to the pot. Season vegetables with salt and pepper and saute until they begin to soften 5 to 7 minutes. Add beans, stock and tomato sauce. Bring soup to a boil over medium high heat. Remove lid and stir in torn stale bread. Stir soup and incorporate bread as it breaks down. When soup is becomes thick and bread is distributed evenly, adjust seasoning and serve the soup in shallow bowls. Some ribolittas are so thick, the spoon can stand upright. This is a subjective technique. Make yours as thick or thin, as you like with the addition of either more bread or additional stock or water.
Top shallow bowlfuls of soup with thinly sliced or finely chopped raw onion, a drizzle of extra- virgin olive oil and a generous sprinkling of grated Parmigiano-Reggiano.