Preheat the oven to 325°F (165°C)
Pat the lamb shanks dry with paper towels
Season the lamb shanks generously with salt and black pepper
Heat oil in a large oven-safe pot or Dutch oven over medium-high heat
Brown the lamb shanks on all sides, then remove them from the pot
Add chopped onions, carrots, and celery to the pot
Cook until softened
Add minced garlic and cook briefly
Stir in tomato paste and cook for 1 to 2 minutes
Pour in red wine or broth and scrape up the browned bits from the bottom
Add broth, herbs such as rosemary and thyme, and bay leaves
Return the lamb shanks to the pot
Bring the liquid to a simmer
Cover the pot with a lid
Transfer to the oven
Braise for 2.5 to 3.5 hours until the meat is very tender
Check occasionally and add more broth if needed
Remove the lamb shanks from the pot
Strain or reduce the sauce if desired
Serve the lamb shanks with the sauce spooned over the top