Put a large pot of water on to boil over high heat. Bring a second large saucepan of water to boil. Add the broccoli rabe to the saucepan and cook until crisp-tender, 2 minutes.
Remove from the heat, drain, and plunge the broccoli rabe into a large bowl of cold water. Transfer to a salad spinner and spin dry. Set aside.
Heat 3 tablespoons of the olive oil in a shallow Dutch oven, over medium-high heat. Add the sausage, breaking it up with a wooden spoon and stirring often, until the sausage is in small pieces and is thoroughly cooked through, about 5 minutes. Reduce the heat to medium-low and stir in the garlic.
Meanwhile, add the cavatelli to the pot of boiling water and cook according to the package directions, 5 to 7 minutes.
Add the crushed tomatoes and the salt to the Dutch oven and cook, stirring occasionally, for 5 minutes.
Drain the cavatelli and add to the Dutch oven with the broccoli rabe and the remaining 3 tablespoons of olive oil, tossing gently to evenly coat. Cook for 2 minutes more.
Serve immediately with the grated Parmigiano-Reggiano cheese and black pepper to taste.