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 Dutch oven over medium until foamy. Add 2 lamb shanks, unpeeled garlic cloves, bay leaves, and thyme. Cook, turning occasionally, until the lamb is browned, about 15 minutes. Transfer to a platter and set aside. Add 1 tablespoon butter and remaining 2 shanks to the Dutch oven; cook until browned, about 15 minutes. Return all to the Dutch oven. Cover and cook in the oven, flipping shanks every 20 minutes, until very tender, 2 to 2.5 hours.

Remove the shanks, pour in chicken stock, and bring to a boil, scraping up browned bits. Remove from heat, skim off fat, and return the stock to a boil until reduced to 2 cups, about 5 to 8 minutes. Strain the stock into the Dutch oven.

Add peeled garlic to the stock and simmer over medium-low until slightly tender, about 20 minutes. Stir in bell peppers and thyme leaves and return the lamb to the Dutch oven. The lamb can be made ahead; cover and refrigerate then reheat before proceeding.

Cook over medium-low, moving peppers and lamb occasionally, until tender, about 10 minutes. Serve the lamb shanks with sauce, peppers, and garlic.