In a saucepan, heat the extra-virgin olive oil. Add the garlic, celery, carrot, and onion and saute until they are soft. Add the tripe, and cook together on medium heat, for about 5 minutes. Add the tomatoes including the juices. Break up the whole plum tomatoes with a fork.
Let the mixture cook together until the sauce has reduced and thickened, about 15 minutes. Remove the pan from the heat, season with salt and pepper, to taste, and add the freshly grated Parmigiano cheese. Stir well and serve warm.
*Tripe must be boiled for approximately 2 hours before use in this recipe.