How to Make It
Preheat oven to 300°F. Sprinkle lamb with 1 teaspoon salt and 1/2 teaspoon black pepper.
Heat 1 tablespoon butter in a large ovenproof Dutch oven over medium until foamy. Add 2 foreshanks, all of the unpeeled garlic cloves, bay leaves, and thyme sprigs, and cook, turning occasionally, until browned all over, about 15 minutes. Transfer browned foreshanks, unpeeled garlic, bay leaves, and thyme sprigs to a platter, and set aside. Add remaining 2 foreshanks to Dutch oven, and cook, turning occasionally, until browned all over, about 15 minutes. Return browned foreshanks, unpeeled garlic, bay leaves, and thyme sprigs to Dutch oven. Cover and transfer to preheated oven. Cook, flipping shanks every 20 minutes, until very tender, about 2 hours.
Remove lamb shanks, and set aside. Pour chicken stock into Dutch oven, and bring to a boil over high, scraping up any browned bits from bottom of Dutch oven. Remove from heat; use a ladle to skim off fat from surface, and discard. Return stock to a boil over high, skimming surface often and discarding fat, until reduced to 2 cups, about 10 minutes. Pour stock mixture through a fine wire-mesh strainer into a large measuring cup; discard solids. Skim remaining fat from surface, and discard. Wipe Dutch oven clean; pour strained stock into Dutch oven.
Add peeled garlic cloves to strained stock, and simmer over medium-low until garlic is slightly tender, about 20 minutes, flipping garlic cloves after 10 minutes. Return lamb to Dutch oven. (The lamb can be cooked up to a day ahead. Refrigerate, covered; reheat before proceeding.)
Scatter bell peppers and thyme leaves around lamb, and cook over medium-low, moving peppers around lamb occasionally, until peppers and garlic are tender, about 10 minutes. Using a slotted spoon, transfer lamb shanks to warm serving plates. Add remaining 1 tablespoon butter, remaining 1/4 teaspoon salt, and remaining 1/2 teaspoon black pepper to sauce, and stir until creamy. Remove from heat. Spoon sauce with bell peppers and garlic cloves around lamb shanks.