Preheat the oven to 325 degrees F. Heat the olive oil in a Dutch oven over medium heat and brown the lamb chunks on all sides. Remove the lamb and set it aside.
Add the onion, carrots, and garlic to the same pot. Cook for 5 minutes until the onion softens.
Stir in the tomato paste, thyme, bay leaves, salt, and pepper. Return the lamb to the pot and pour in the beef broth. Bring the mixture to a simmer.
Cover the Dutch oven and transfer it to the oven. Braise for 2 hours until the lamb is fork-tender.
While the stew braises, place the potatoes in a large pot and cover with water. Boil until soft, about 20 minutes. Drain the potatoes.
In a skillet, melt 2 tablespoons of the butter over medium heat. Add the shredded cabbage and cook until wilted, about 8 minutes.
Mash the drained potatoes with the remaining 2 tablespoons of butter and the milk. Stir in the cooked cabbage and season with salt and pepper.
Remove the stew from the oven and discard the bay leaves. Serve the stew alongside the colcannon.