In a large pot combine the tripe, vinegar, vanilla and enough water to cover the tripe by 2 inches. Bring to a boil, reduce to a simmer and cook until the tripe is very tender, about 1 to 1 1/4 hours, replenishing the water as necessary.
Drain the tripe, reserving the cooking liquid, and allow to cool. Slice the tripe into 1-inch strips.
In a 14 to 16-inch skillet, heat the olive oil over high heat until almost smoking. Add the onion, garlic and tripe and saute 3 minutes. Add the tomato sauce, bring to a boil, reduce to a simmer and cook, covered, for 30 minutes.
Meanwhile, in a small bowl, combine the grated cheeses and the mint and stir to combine. When the tripe is finished, divide evenly among 6 warmed bowls and top with the cheese and mint mixture.
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds 1 week in the refrigerator or up to 6 months in the freezer.
Recipe copyright 2000, Mario Batali. All Rights Reserved