Preheat the oven to 300 degrees F. Combine the chopped vegetables, garlic and thyme in a large roasting pan. Rub the dried herbs and spices all over the beef; season generously with salt and pepper. Place the beef fat-side up on the vegetables and roast until a thermometer inserted into the thickest part of the meat registers 130, about 45 minutes. Remove the meat from the pan, cover loosely with foil and set aside to rest for 30 minutes.
While the meat is resting, put the roasting pan on the stove and cook the vegetables over high heat until they are browned but not burnt and the liquid has evaporated; pour off any excess fat. Add the beef broth to the pan, using a wooden spoon to scrape up any browned bits that cling to the bottom. Transfer the liquid and vegetables to a medium saucepan and bring to a boil. Reduce the heat to low and simmer until the liquid reduces by half, about 30 minutes. Strain the sauce and season with salt and pepper. Keep warm.
Trim any excess fat from the beef and thinly slice. Halve the rolls horizontally and warm in the oven. Dip the bottom of each roll in the sauce and top with some of the beef; dip the cut side of each roll top in the sauce and place on each sandwich. Serve with the hot mustard.
Recipe courtesy Phillipe the Original, Los Angeles, CA for Food Network Magazine