Heat 3 tablespoon of the olive oil in a deep 12-inch skillet or a large Dutch oven set over medium heat. Add the onion, stirring until it softens, about 10 minutes. Add the sausage, stirring to break into bite-sized pieces and beginning to brown on all sides, about 5 minutes.
Meanwhile, as the onion and then sausage cooks, set up your mise en place for the rest of the dish. Dice your vegetables, mince your garlic, and chop your herbs. Grate the cheese (use a food processor to quickly grate hunks of Parmigiano).
Stir the diced carrot and celery into the pan and continue cooking another 20 minutes, until the onion is well browned and the vegetables are tender. If the pan seems dry, add a little more of the olive oil to keep things from sticking.
Stir in the herbs, garlic, and chile flakes, and cook until they become fragrant, about 2 minutes. Stir in the tomato paste and let it brown, about 1-2 minutes more.
Stir in the whole plum tomatoes together with their juice and the 1 ½ cups water. Break up the tomatoes with your stirring spoon. Mix in the lentils, the 1 ½ teaspoons salt, and the ½ teaspoon pepper. Bring the sauce to a simmer, then cook gently over low heat, until the lentils are tender (about 40-60 minutes, but this depends significantly on the source and age of your lentils). If the sauce begins to dry out towards the end of cooking, add a little more water, chicken broth, or tomato sauce.
Towards the end of the lentil cooking time, bring a large pot of heavily salted water to a boil. Add the pasta and cook until it is just al dente, about 1 minute less than the package/recipe guidance.
Scoop out some of the pasta cooking water, using a coffee mug or measuring cup, and reserve it for later.
Drain the pasta and transfer the pasta to a large serving bowl. Toss the pasta with the 2 tablespoons butter (if desired), add a generous splash of the cooking water, the lentil-sausage sauce, and the cheese. Optionally, drizzle with additional olive oil, garnish with chives or parsley, and serve with fresh black pepper and more Parmigiano-Reggiano on the side.