1. Tomato Sauce: Heat olive oil in a 3-quart saucepan over medium heat.
2. Add onion and garlic; cook until soft and light golden brown, 8 to 10 minutes.
3. Add thyme and carrots; cook 5 minutes more, until carrot is quite soft.
4. Add tomatoes with their juice and bring to a boil, stirring often.
5. Lower the heat and simmer for 30 minutes until as thick as hot cereal.
6. Season with salt to taste.
7. Add tomato sauce to a medium saucepan and bring to a boil.
8. Add the sausage to the sauce and reduce the heat to a simmer.
9. Cook for 1 hour, skimming off the fat as it is rendered from the meat.
10. Remove sauce from the heat, cool briefly, and then pulse it in a food processor until smooth.
11. Transfer to a 12-inch pan and keep warm.
12. Bring 5 litres of water to a boil and add 2 tablespoons of salt (optional).
13. Cook tortiglioni according to package directions.
14. Drain tortiglioni and add to the pan with the sauce.
15. Toss gently over high heat for 1 minute.
16. Add mint leaves and toss for 1 more minute.
17. Divide evenly among warmed dinner plates.
18. Grate Pecorino Romano cheese over each plate and serve immediately.