Preheat your oven to 325°F (165°C). Pat each lamb shank completely dry with paper towels, then season all sides generously with kosher salt and black pepper, pressing the seasoning into the meat with your hands.
Heat 2 tablespoons of olive oil in a Dutch oven over medium-high heat until shimmering. Sear the lamb shanks in batches for 10–12 minutes per batch, turning every 3–4 minutes, until a deep golden-brown crust forms on all sides, then transfer to a plate.
Reduce heat to medium, add the remaining 1 tablespoon of olive oil, and cook the onion, carrot, and celery for 8–10 minutes until softened. Stir in the minced garlic for 1–2 minutes, then add the tomato paste to the center of the pot and cook, stirring constantly, for 2 minutes until darkened and fragrant.
Pour in the red wine and scrape up all the browned bits from the bottom of the pot, then bring to a vigorous simmer and cook for 5–6 minutes until reduced by half. Add the beef stock and crushed tomatoes with their juices, then nestle in the rosemary, thyme, and bay leaves.
Return the seared lamb shanks and any collected juices to the pot, nestling them into the braising liquid. Cover with the lid, transfer to the preheated oven, and braise for 2 to 2.5 hours, turning the shanks once at the halfway point, until the meat is completely fork-tender and pulling away from the bone.
Remove the shanks and set aside tented with foil; discard the herb sprigs and bay leaves, then simmer the braising liquid over medium-high heat for 10–15 minutes until reduced to a glossy, thick sauce. Return the shanks to the pot, spoon the sauce over them generously, and serve garnished with chopped fresh parsley.