Season lamb shanks with salt and pepper.

Heat olive oil in a deep-sided pan or pot over medium-high heat. Add lamb and cook, turning as needed, until browned, about 5 minutes. Remove from pan and reduce heat to medium.

Place chopped onion, carrot, celery, and garlic in the pan. Season with a large pinch of salt. Cook and stir until vegetable juices start to come out, about 2 minutes. Add tomato paste and stir to coat, about 1 minute. Pour in beer and stir in rosemary. Raise heat to high and bring sauce mixture to a simmer.

Toss lamb shanks in with the sauce mixture. Reduce heat to low. Cover and simmer until lamb is nearly fork-tender, flipping lamb after 1 hour, about 2 hours. Remove from heat and let cool to room temperature, at least 30 minutes. Refrigerate, 8 hours to overnight.

Skim the fat off the top of the sauce, if desired. Cover and bring lamb to a simmer over low heat. Flip and continue simmering until meat is fork-tender and nearly falling off the bone, about 45 minutes. Place lamb shanks in a bowl to keep warm.

Bring sauce to a boil over high heat. Boil until reduced by half but not too thick, about 3 minutes. Season with salt and cayenne pepper. Plate the lamb shanks and spoon sauce on top.