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