I made a roast beef the other night that was flavorful, but a bit hard. Would I be able to throw it in a Dutch oven with stock to braise it further and use it as shredded beef for a different meal?
Can't see why not. For best flavor, use beef or veal stock (which after use in braising may be converted to gravy or soup), and herbs/spices (if any) similar to what you used in roasting. Cut in slices or chunks before braising for more flavor & quicker results. Braise covered.
If it's a lean cut of beef - which given the texture, sounds possible - you might want to brown some bacon, pancetta or such before adding the meat and stock. A good braise benefits from a little fat.
I used some leftover bacon to start the braising liquid, and braised covered, in chunks, as per Nancy's suggestion above. We had the shredded beef this evening for dinner as a french dip, and it was delicious! The bacon gave the liquid a wonderful flavor, and the dish really worked well. Thanks!
Braising lean meat is never a good idea. It will wind up with the consistency of sawdust. Especially if it has already been roasted. I would not advise you do that