Adjust oven rack to middle position and heat oven to 300 degrees. Combine beef, broth, water, chopped carrots, celery, onion, bay leaves, peppercorns, thyme, and allspice in Dutch oven. Cover and bake until fork slips easily in and out of meat, 4 1/2 to 5 hours.

Transfer meat to 9x13 inch baking dish. Strain cooking liquid through fine-mesh strainer into large bowl, discard solids, and skim fat from liquid. Pour 1 cup cooking liquid over meat. Cover dish tightly with aluminum foil and let rest for 30 minutes.

Meanwhile, return remaining cooking liquid to Dutch oven, add butter, and bring to simmer over medium-high heat. Add potatoes and simmer until they begin to soften, about 10 minutes. Add carrot halves, cover, and cook until tender, 10 to 15 minutes. Transfer vegetables to serving platter and season with pepper to taste.

Transfer beef to carving board and slice against grain into 1/4-inch-thick slices. Serve with vegetables.