Bring a large pot of salted water to a boil for the pasta.
Heat a large Dutch oven over medium heat, and add the olive oil. When the oil is hot, crumble in the sausage. Crumbling it with a wooden spoon, cook until the sausage is browned, about 5 minutes.
Add the red onion and thyme. Stir and cook until the onion begins to wilt, about 4 minutes.
Stir in the cabbage, and season with 1 teaspoon salt and a pinch of peperoncino. Cook until the cabbage begins to brown and wilt, 5 to 6 minutes.
Make a space in the pan, and add the tomato paste to that spot. Cook and stir the tomato paste there for a minute, until it darkens a shade or two; then stir it into the cabbage.
Add 2 cups pasta water, and bring the liquid to a simmer. Cover, and cook until the cabbage is very tender, 20 to 25 minutes.
Uncover, and bring the sauce to a rapid simmer to thicken it slightly, if needed, about 2 minutes.
When the cabbage is almost done, add the pasta to the boiling water and cook until it’s al dente.
Add it to the simmering sauce. Toss to coat the pasta with the sauce, adding a little pasta water if it seems dry.
Remove from the heat, sprinkle with the grated cheese, toss, and serve.