Cook pasta according to package directions, decreasing time by 3 minutes. Drain, reserving 2 cups pasta water. Toss pasta with 2 teaspoons oil.
In a 6-qt. stockpot, cook sausage over medium heat until no longer pink, 5-7 minutes, breaking sausage into large crumbles. Add garlic and remaining oil; cook and stir 2 minutes. Stir in kale, salt and pepper; cook, covered, over medium-low heat until kale is tender, about 10 minutes, stirring occasionally.
Add pasta and reserved pasta water; bring to a boil. Reduce heat; simmer, uncovered, until pasta is al dente and liquid is absorbed, about 3 minutes, tossing to combine. Sprinkle with cheese.