Preheat oven to 325°F (160°C).
Heat olive oil in a large oven-safe pot or Dutch oven over medium-high heat. Sear lamb shanks on all sides until browned. Remove and set aside.
In the same pot, sauté onion and garlic until fragrant, about 3–4 minutes. Stir in tomato paste and cook 1 minute.
Add broth, rosemary, salt, and pepper. Return lamb shanks to the pot. Bring to a simmer, then cover and braise in the oven for 2–3 hours, until meat is tender.
Prepare the Mashed Potatoes:
Boil potatoes in salted water until tender, about 15–20 minutes.
Drain, mash with butter and milk or cream. Season with salt and pepper.
Roast the Vegetables:
Toss carrots (and parsnips, if using) with olive oil, salt, and pepper.
Roast in the oven at 400°F (200°C) for 25–30 minutes, until golden and tender.
Serve:
Plate mashed potatoes, top with a braised lamb shank, and arrange roasted vegetables on the side. Spoon some of the braising liquid over the lamb and potatoes for extra flavor.
