Heat oil in a 12-inch heavy skillet over moderately high heat until hot but not smoking. Add garlic, onion and sausage, breaking up any lumps with a spoon, until browned, 5 to 7 minutes.
Meanwhile, blanch kale in a 6-quart pot of boiling salted water, uncovered, 5 minutes. Remove kale with a large sieve and drain. Return cooking water in pot to a boil, then cook pasta in boiling water, uncovered, until al dente. Reserve 1 cup pasta-cooking water, then drain pasta in a colander.
While pasta cooks, add kale to sausage in skillet and sauté, stirring frequently, until just tender, about 5 minutes. Add broth, stirring and scraping up any brown bits from bottom of skillet, then add pasta and 1/2 cup reserved cooking water to skillet, tossing until combined. Stir in cheese and thin with additional cooking water if desired.
Serve immediately, with additional cheese on the side.