In a very large pot, bring 5 quarts water and 3 tablespoons salt to a boil. Add the pasta and cook until just al dente, about 1 minute less than the package directions. Occasionally give the pasta a stir so it doesn’t stick together. Scoop out and reserve 1 cup of the pasta water before draining the pasta.
Meanwhile, put large skillet over medium-high heat. Add the olive oil. When the oil is hot, add the sausage and cook, stirring only occasionally, until very well browned, about 4 minutes. Add the kale and cook for 1 minute, stirring constantly.
Add the reserved pasta water and deglaze the pan, scraping with a wooden spoon to get up the browned bits on the bottom of the pan. Add the cooked pasta to the pan, stirring the noodles into the sauce.
Remove from the heat and stir in the Parmesan and extra-virgin olive oil. Taste and adjust the seasoning adding additional Parmesan if desired.