PLACE meat in deep roasting pan. Combine broth, water, steak sauce, vegetables, garlic and seasonings; pour over meat. Cover. Bake at 375ºF for 4 hours or until meat is medium doneness (160ºF).
REMOVE meat from braising liquid; cool. Strain braising liquid; discard strained vegetables. Pour about 1-1/4 gal. (or 2-1/2 qt. for trial recipe) of the strained liquid into large saucepan; bring just to boil. Simmer 15 to 20 min. or until slightly thickened. Remove 2 qt. of the reduced liquid (or 1 qt. of the reduced liquid for trial recipe); pour into separate saucepan to keep warm to serve with the sandwiches.
SHRED meat with fork. Add to remaining liquid in saucepan; cook until heated through, stirring occasionally.
FOR each serving:
Toast cut sides of 1 of the rolls. Use a slotted spoon to place about 3/4 cup of the meat in roll; top with 1 cheese slice. Serve with 1/4 cup of the reserved liquid in small ramekin.