01 - Preheat oven to 320°F or set slow cooker to low setting. Pat lamb shanks dry with paper towels and season generously with salt and pepper on all sides.
02 - Heat olive oil in large Dutch oven over medium-high heat. Sear lamb shanks on all sides until deeply browned, approximately 8 minutes total. Transfer to a plate.
03 - In the same pot, add onions, carrots, and celery. Cook for 5-7 minutes until softened, stirring occasionally. Add minced garlic and cook for 1 minute longer.
04 - Stir in tomato paste and cook for 1 minute. Pour in red wine, scraping up browned bits from pot bottom. Simmer for 2-3 minutes to reduce slightly.
05 - Add beef stock, rosemary sprigs, thyme, and bay leaves. Return lamb shanks to pot, nestling them into vegetables. Bring to gentle simmer.
06 - Cover pot and transfer to oven. Braise for 3 to 3.5 hours at 320°F until meat is very tender and nearly falling from bone. Alternatively, transfer all contents to slow cooker and cook on low for 8 hours.
07 - Remove lamb shanks to warm serving platter and tent with foil. Discard bay leaves and rosemary stems from braising liquid.
08 - Skim excess fat from surface of sauce. If thicker gravy is desired, whisk cornstarch with 2 tablespoons cold water until smooth, then slowly whisk into simmering sauce until thickened. Taste and adjust seasoning with salt and pepper.
09 - Arrange lamb shanks on serving plates. Ladle rosemary-infused gravy generously over meat and accompanying vegetables.