Preheat the oven to 300°F. Sprinkle the lamb with salt and 1/2 teaspoon black pepper.

Heat 1 tablespoon of the butter in a large ovenproof Dutch oven over medium until foamy. Add 2 of the lamb foreshanks, all of the unpeeled garlic cloves, bay leaves, and thyme sprigs, and cook, turning occasionally, until the lamb is browned all over, about 15 minutes. Transfer the browned foreshanks, unpeeled garlic, bay leaves, and thyme sprigs to a platter, and set aside. Add 1 tablespoon of the butter and the remaining 2 foreshanks to the Dutch oven, and cook, turning occasionally, until browned all over, about 15 minutes, reducing the heat as needed to prevent over-browning. Return the browned foreshanks, unpeeled garlic, bay leaves, and thyme sprigs to the Dutch oven. Cover the Dutch oven and transfer it to the preheated oven. Cook, flipping shanks every 20 minutes, until very tender, 2 hours to 2 1/2 hours.

Remove the lamb shanks from the Dutch oven, and set aside. Pour chicken stock into the Dutch oven, and bring to a boil over high, scraping up any browned bits from bottom of the Dutch oven. Remove it from heat; use a ladle to skim off fat from surface, and discard. Return the stock to a boil over high, stirring occasionally, until reduced to 2 cups, 5 to 8 minutes. Pour the stock mixture through a fine wire-mesh strainer into a large measuring cup; discard solids. Wipe the Dutch oven clean and pour the strained stock into the Dutch oven.

Add the peeled garlic cloves to the strained stock, and gently simmer over medium-low until the garlic is slightly tender, about 20 minutes, flipping the garlic cloves after 10 minutes. Stir in the bell peppers and thyme leaves and return the lamb to Dutch oven. (The lamb can be cooked up to a day ahead. Refrigerate, covered; reheat before proceeding.)

Cook over medium-low, moving peppers and lamb occasionally, until the peppers and garlic are tender, about 10 minutes. Remove from heat. Using a slotted spoon, transfer the lamb shanks to warm serving plates. Add the remaining 1 tablespoon butter and remaining 1/2 teaspoon black pepper to sauce, and stir until creamy. Season with additional salt to taste. Spoon the sauce with bell peppers and garlic cloves around the lamb shanks and serve.