In a food processor, pulse the carrots, celery, and onions until finely chopped. In a 12-inch skillet, heat 2 tablespoons olive oil over low. Add the vegetables, and cook, stirring, until tender and lightly caramelized, about 1 hour.
Meanwhile, in a large pot of boiling water, add the sweetbreads and cook until tender, about 15 minutes. Drain the sweetbreads, and then let cool to room temperature. Using a paring knife, remove and discard the outer membrane, and cut the sweetbreads into 1⁄2-inch cubes.
In a large saucepan of cold, salted water, add the tripe and bring to a boil. Drain the tripe and then repeat this process twice, allowing the tripe to sit in the boiling water for 15 minutes after the third time boiling it. Drain the tripe once more, let cool to room temperature, and then cut into 1⁄2-inch cubes.
In a 12-inch skillet, add 1 cup of the guanciale, and then heat over medium-high. Cook, stirring, until golden brown and crisp, about 6 minutes. Using a slotted spoon, transfer the guanciale to paper towels to drain and return the skillet with its fat to medium heat. Add the tomato paste and cook, stirring, until slightly caramelized, about 1 minute. Add the caramelized vegetables, and cook, stirring, for 2 minutes. Add the wine, and cook, stirring, until reduced by half, about 2 minutes. Add the reserved tripe along with the tomato purée and 1⁄2 cup water and bring to a boil. Reduce the heat to low, and cook, covered and stirring occasionally, until the tripe is tender, about 2 1⁄2 hours.
Add the sweetbreads and 1⁄2 cup water to the sauce, and cook, stirring, until the sweetbreads break down into the sauce, about 20 minutes. Stir the cooked guanciale back into the sauce, and season to taste with salt and pepper. Remove the sauce from the heat and let cool to room temperature. (The ragu can be made up to this point and refrigerated for 3 to 5 days or frozen for 3 to 4 weeks.)
In a large pot of boiling, salted water, add the rigatoni and cook until al dente, about 13 minutes. Meanwhile, place the remaining 1⁄4 cup guanciale in a 12-inch skillet and then heat over medium-high. Cook, stirring, until the guanciale is caramelized and crisp, about 6 minutes. Add the remaining 1 tablespoon olive oil and the chile flakes, and cook for 1 minute. Using tongs, transfer the rigatoni to the sauce along with 1 cup of pasta water and toss to combine.
In a small bowl, stir the ricotta with the mint, and then divide the ricotta among 4 pasta bowls. Spoon the pasta over the ricotta in each bowl and then top with more pecorino to serve.