Cavatelli with Sparerib Ragù

Ingredients

• One 4-pound rack of pork spareribs, rack cut crosswise through the rib bones into 3 strips (have your butcher do this)
• 1/4 cup plus 1 tablespoon extra-virgin olive oil
• Kosher salt
• pepper
• 1 large yellow onion, finely chopped
• 5 carrots, finely chopped
• 4 celery ribs, finely chopped
• 4 garlic cloves, crushed
• 2 tablespoons chopped thyme
• 1/2 pound pork skin, cut into small dice
• 2 cups dry red wine
• One 28-ounce can crushed tomatoes
• 2 pounds cavatelli
• 1 cup freshly grated Parmigiano-Reggiano cheese, plus more for serving