Preheat oven to 325°F (165°C).
Pat lamb shanks dry and season generously with salt and black pepper.
In a heavy pot or Dutch oven, heat olive oil over medium-high heat. Sear lamb shanks on all sides until golden brown, then remove and set aside.
In the same pot, sauté onion, carrots, and celery for 5–7 minutes until softened and lightly golden.
Add garlic and rosemary, cooking briefly until fragrant.
Pour in additional beef stock (in place of wine), scraping up browned bits from the bottom of the pot.
Return lamb shanks to the pot and add remaining beef stock until mostly submerged.
Bring to a gentle simmer, cover, and transfer to the oven.
Bake for 2 1/2 to 3 hours, turning once halfway through, until the meat is tender and nearly falling off the bone.
Remove lamb shanks and keep warm. Strain the cooking liquid into a saucepan.
Simmer the liquid until reduced by half.
Remove from heat and whisk in butter, chopped rosemary, and lemon juice. Adjust seasoning as needed.
Serve lamb shanks with the rosemary jus spooned over the top.