About 3/4 cup freshly grated Parmigiano-Reggiano
6 tablespoons butter
Prepare a soffritto and braise the rabbit. Combine the onion, celery and carrot in a food processor and pulse until finely chopped. (Alternatively mince the vegetables my hand.)
Preheat the oven to 350. Heat the oil in a large high-sided skillet or Dutch oven over medium-high heat.
Season the rabbit pieces on both sides with salt and pepper. Working in batches if necessary to avoid crowding the pan, brown the rabbit, about 5 minutes per side. Remove the rabbit from the pan and reserve.
Reduce the heat to medium. If the pan looks dry add a little more oil. Add the finely chopped soffritto vegetables. Season with a little salt and pepper, and cook, stirring constantly at the beginning then frequently as the vegetables soften and brown slightly, about 10 minutes.
Stir in the tomato paste. Make sure the paste is well distributed and the aromatic vegetables are well covered. Cook, stirring often, for 5 minutes. Break the tomatoes into the pot with your hands. Raise the heat to medium-high and cook until the tomato juices concentrate and evaporate, about 5 minutes. Add the olives and wine and simmer until the pan is almost dry, just a few minutes. Return the rabbit to the pan. Pull the leaves from the rosemary branch and add them to the pan.
Add enough stock to surround but not cover the rabbit, it should come up about two-thirds up the sides of each piece. Bring the stock to a simmer on top of the stove then cover the pot and braise in the oven for 20 minutes. Baste the rabbit then recover and braise for 20 minutes more. Baste the again, then braise, uncovered, basting every 10 minutes until the rabbit is tender, about 20 minutes more. (Note the braised rabbit is very good simply served as is.) Remove the pot from the heat and allow the rabbit to cool. Remove the meat from the bones and return it to the pan. Discard the bones. (The ragu can be prepared ahead and refrigerated or frozen.)
Bring a large pot of salted water to a boil.
Warm the rabbit ragu over medium-low heat. Add a little more stock or water, enough so it looks a little soupy. Add the pasta to the boiling water and cook until it flexible but not yet al dente, about 3 minutes for fresh pasta, longer for dried.
Stir ½ cup of cheese and 4 tablespoons of butter into the ragu. Using a slotted spoon, add the pasta to the sauce (along with a little of the pasta cooking water). Let the pasta simmer until it is al dente, about 2 minutes.
Season the pasta and sauce with salt and pepper and toss it gently. Add the remaining 2 tablespoons of butter then divide the paparadelle between warm bowls. Serve topped with the reserved cheese and a bit more fresh pepper.