Preheat the oven to 300°F. Place a large Dutch oven or braiser over medium high heat. Add the olive oil and begin browning the lamb shanks, a few at a time. Sear the shanks until brown on several sides. Turn the heat down to medium low, adding a bit more olive oil if needed.
Add the diced onions and sliced leeks to the Dutch oven. Sauté for 3 to 5 minutes until the onions and leeks are softened. Add the minced garlic and sauté for another minute until fragrant. Stir the tomato paste into the onion mixture and then add the amber ale.
Bring the ale to a simmer and then add the beef broth, fresh thyme bundle, bay leaves, brown sugar, kosher salt, and pepper. Return the shanks to the Dutch oven. Cover and place in the oven to braise.
Braise the lamb shanks for around 1 hour and 45 minutes. Remove from the oven and add the sliced carrots and rinsed barley, stirring them into the liquid. Cover and return to the oven for enough 30 to 45 minutes or until the barley is fully cooked.
The lamb shanks are done when they are very tender and falling off of the bone. Remove the lamb shanks to a serving platter. Taste the barley stew and adjust the seasoning if needed. Stir in the parsley. Serve the lamb shanks with the barley stew and enjoy!