Heat oil in a Dutch oven or large, wide pot over medium-high heat until oil begins to smoke. Sear lamb shanks in hot oil on all sides until browned, about 10 minutes. Remove lamb shanks and set aside. Drain excess grease.

Add onion and garlic to the Dutch oven over medium heat. Cook and stir until onion is softened and translucent, about 5 minutes. Stir in carrots, celery, and tomato paste; cook for 5 minutes more.

Return lamb shanks to the Dutch oven and pour in broth and stout beer. Use kitchen twine to tie together thyme, parsley, and bay leaf into a secure bundle; add to lamb shanks. Bring to a boil.

Reduce heat to medium-low, cover, and simmer, stirring occasionally, until lamb is tender and nearly falls off the bone, 2 to 3 hours. Stir in rosemary sprig and season with salt and pepper during the last 10 minutes of cooking. Remove rosemary sprig and herb bundle before serving.