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Season the lamb shanks: Pat the lamb shanks dry with paper towels and season generously with salt and pepper.
Brown the lamb shanks: In a large heavy-based pot, heat 2 tablespoons of olive oil over high heat. Sear the lamb shanks in batches until browned all over, about 5 minutes per batch. Transfer the lamb to a plate and drain any excess fat from the pot if necessary.
Saute the aromatics: Reduce the heat to medium-low and add the remaining 1 tablespoon of olive oil to the pot. Add the chopped onion and minced garlic, cooking for about 2 minutes until fragrant. Then, add the chopped carrots and celery, cooking for another 5 minutes until the onion becomes translucent.
Prepare the braising liquid: Pour in the red wine, chicken stock, crushed tomatoes, and tomato paste. Add the sprigs of thyme and bay leaves, stirring to combine.
Add the lamb shanks: Return the lamb shanks to the pot, arranging them so they are mostly submerged in the liquid.
Stove braising (2 hours covered): Increase the stove heat to bring the liquid to a simmer. Cover the pot with a lid and transfer it to the preheated oven. Cook for 2 hours.
Finish uncovered (30 minutes): Remove the lid and continue cooking in the oven for an additional 30 minutes. The meat should be ultra tender but still clinging to the bone. If not, cover and continue cooking until desired tenderness is achieved.
Prepare the sauce: Remove the lamb shanks from the pot and keep warm. Discard the bay leaves and thyme sprigs. Use a hand blender to blitz the vegetables into the sauce. If needed, simmer sauce over medium heat until the sauce reaches a slightly syrupy consistency. Taste and adjust seasoning with salt and pepper as needed.
Serve: Serve the lamb shanks over mashed potatoes or cauliflower puree, generously spooning the sauce over the top. Garnish with fresh thyme leaves if desired.
