Prepare the Sauce

1. Heat a large saucepan over medium heat. Add olive oil and sweat the diced onion, garlic, thyme, and bay leaf until soft and translucent, about 5 minutes. Do not brown them.
2. Stir in the tomato paste and cook for an additional 2-3 minutes until it deepens in color.
3. Pour in the white wine and reduce by two-thirds. Add the tomato puree and simmer over medium heat, stirring occasionally for about 30 minutes, until thickened.
4. Season with salt and pepper, then stir in the olive oil. Let it cool slightly before refrigerating until needed.

Cook the Lamb

1. Set the oven to 325°F (165°C).
2. In a large oven-safe pot or Dutch oven, heat olive oil over medium-high heat. Season the lamb shanks generously with salt and pepper. Sear them until browned on all sides.
3. Add bay leaves, garlic, onion, and carrot to the pot. Pour in chicken stock, ensuring the lamb is mostly submerged.
4. Cover tightly with foil or a lid and place in the oven. Braise for 2 1/2 to 3 hours until the lamb is tender and falling off the bone.
5. Remove from the oven and let the lamb cool in the cooking liquid. Once cool, shred the meat and discard the bones.
6. Strain the braising liquid and mix it into the tomato sauce. Simmer the combined sauce over medium-low heat for another 15-20 minutes to let the flavors meld.

Cook the Pasta

1. Cook the pasta in a large pot of salted water according to the package instructions, until al dente. Drain and reserve a bit of pasta water.
2. Toss the pasta with the lamb ragu sauce. If the sauce is too thick, add a little pasta water to loosen it to your desired consistency.

Assemble and Serve

1. Plate the pasta and lamb ragu, garnishing with fresh basil leaves and shaved Parmesan cheese.