Cook the Pancetta: Heat the olive oil in a large Dutch oven or heavy-bottomed pot over medium heat. Add the pancetta and the whole dried chile, if using. Cook for 4 to 5 minutes, stirring occasionally, until the pancetta begins to brown and release its fat.
2 tablespoons extra-virgin olive oil, 3½ ounces pancetta, 1 small dried red chile or ¼ teaspoon red pepper flakes
Cook the Vegetables: Add the onion, carrot, celery, salt, and black pepper. Cook for 8 to 10 minutes, stirring frequently, until the vegetables are very soft but not browned. Add 1 tablespoon of water at a time if the vegetables begin to stick or darken.
1 medium yellow onion, 1 medium carrot, 1 celery stalk, ½ teaspoon kosher salt, ¼ teaspoon ground black pepper
Add the Tripe: Stir in the fully cooked tripe. If using red pepper flakes instead of a whole dried chile, add them at this stage so they do not burn. Cook for 5 minutes, stirring occasionally, to help the tripe absorb the flavors of the pancetta and vegetables.
2 pounds fully cooked tripe
Deglaze with Wine: Pour in the white wine and scrape any browned bits from the bottom of the pot. Cook for 2 to 3 minutes, or until most of the wine has evaporated.
¼ cup dry white wine
Add the Tomatoes: Stir in the tomato passata or crushed tomatoes. Bring the sauce to a gentle simmer, then reduce the heat to low and partially cover the pot.
1¼ cups tomato passata or crushed tomatoes
Simmer the Tripe: Cook for 35 to 45 minutes, stirring occasionally, until the sauce has thickened and the tripe is tender but still slightly chewy. Add a small amount of water if the sauce becomes too thick before the tripe is ready.
2 to 4 tablespoons water
Add the Mint: Stir in the chopped mint during the final 5 minutes of cooking. Taste the sauce and add more salt or black pepper as needed.
⅓ cup loosely packed fresh mint leaves
Finish and Serve: Remove and discard the whole dried chile, if used. Stir in the Pecorino Romano, then spoon the tripe and tomato sauce into bowls. Serve hot with additional grated Pecorino Romano.
½ cup freshly grated Pecorino Romano cheese