Bring a large pot of salted water to a boil. Heat a large skillet over medium heat and add the olive oil. Crumble the sausage into the skillet and add the onion. Cook, breaking up the meat with a wooden spoon, until the onion is softened, about 8 minutes. Increase the heat to medium high and cook, stirring, until the sausage and onion are lightly browned, about 2 minutes.
Push the mixture to the sides of the skillet and add the tomato paste in the middle. Cook the tomato paste, stirring, until darkened, 1 to 2 minutes. Add the rosemary, then stir the tomato paste into the sausage and onion. Stir in the wine and bring to a simmer. Cook until the wine is reduced by half, about 2 minutes. Add 1 1/2 cups of the boiling water and bring the sauce to a boil. Cook until thickened, about 5 minutes.
While the sauce cooks, add the cavatelli and peas to the remaining boiling water and cook as the cavatelli label directs. Reserve 1/2 cup pasta water, then drain.
Add the cavatelli, peas and parsley to the sauce and bring to a simmer. Toss to coat, adding more cooking water as needed to loosen. Remove from the heat, sprinkle with the cheese and toss. Divide among bowls and top with more cheese.