Bring a large pot of lightly salted water to a boil. Add the pasta and cook until just tender, about 8 to 10 minutes. While the pasta cooks, place the oil in a large skillet set over medium heat. Add the kale, proscuitto, garlic and chili flakes. Cook and stir 4 to 5 minutes, until the kale has softened. Add the stock and bring to a simmer.
When the pasta is cooked, reserve ½ cup of its cooking liquid, then drain the pasta well. Add the pasta, reserved cooking liquid, and 1/3 cup of cheese to the skillet. Season with salt and pepper and toss to combine. Spoon the pasta into bowls and serve with additional Parmesan cheese at the table.