Preheat your oven to 350°F (175°C).
Season the lamb shanks with salt and pepper.
Heat olive oil in a Dutch oven over medium-high heat. Brown the lamb shanks on all sides, then remove and set aside.
In the same pot, add the chopped onion and cook until softened. Add the minced garlic and cook for another minute.
Pour in the beef broth and red wine, scraping up any browned bits from the bottom of the pot.
Return the lamb shanks to the pot and add the rosemary sprigs. Bring to a simmer.
Cover the pot and transfer to the preheated oven. Braise for about 3 hours, or until the lamb is tender and falling off the bone.
Remove the lamb shanks from the pot and keep warm. Skim any excess fat from the surface of the sauce and reduce if necessary.
Serve the lamb shanks with the sauce spooned over the top.