Sauté Sausage and Vegetables: Warm the olive oil in a large skillet over medium heat. Add the sausage and onion, cooking until the onion softens, about 5 minutes. Stir in the carrot and celery, then continue cooking until the vegetables are tender and the sausage is browned, around 20 minutes. Add a little extra olive oil if the skillet gets dry.
Season and Build the Sauce: Add the rosemary, sage, garlic, and chile flakes to the skillet. Stir and cook until fragrant, approximately 2 minutes. Mix in the tomato paste and let it cook for 1 to 2 minutes until it deepens in color.
Simmer the Lentils: Pour in the canned tomatoes along with 1 ½ cups of water. Use a spoon to break the tomatoes into smaller pieces. Stir in the lentils, kosher salt, and black pepper. Bring the mixture to a gentle simmer and let it cook until the lentils are soft, which takes 40 to 60 minutes depending on their age. If the sauce starts to dry out, add more water as needed.
Cook the Pappardelle: Boil a large pot of salted water and cook the pappardelle until it is just shy of fully cooked, about 1 minute less than the package instructions. Reserve a cup of the pasta cooking water before draining.
Combine and Serve: Transfer the drained pasta to the skillet with the lentil-sausage sauce. If desired, stir in butter and a splash of the reserved pasta water to loosen the sauce. Toss everything together and add the Parmigiano-Reggiano cheese, if using. Finish with a drizzle of olive oil and garnish with chopped parsley or chives. Serve immediately, seasoning with additional black pepper to taste.