Barbacoa, unlike what the name would suggest, hasn't much to do with barbecue. It's a traditional weekend treat (presumably as it takes some time to prepare) and is a specialty of northern Mexico in particular. It consists of beef, seasoned with little apart from garlic, cooked long and slow in little liquid until it's reduced to an unctuous beefy mass.
I've made a type of faux barbacoa a number of times using shin of beef and a slow cooker. Shin, in fact any cheap cut of beef, works fine, however in order to cover the shin pieces I had to use a little more water than I would have liked. This meant there was more liquid which had to be reduced off at the end of cooking to reach the right consistency. To make a more traditional barbacoa the best cuts of meat to use are cheek of beef and tongue of beef, however if the thought of cooking tongue of beef doesn't appeal then the cheek alone will do just fine.
Traditionally the meat would be cooked wrapped in maguey leaves and cooked underground. Very tasty I'm sure but not practical for my purposes! I have also seen barbacoa prepared commercially buy enclosing the meat in large torpedoes of aluminum foil which are then cooked slowly, standing upright, in a large pot. However for home cooking the best, and simplest method, is a slow cooker.
If you are using cheek of beef alone (a single cheek of beef is about the size of a small chicken) then the beef can be cut into inch cubes added to the slow cooker along with a couple of cloves of garlic, some salt and a small amount (a cup or two) of water. If you are using both cheek and tongue then you will to arrange things in such a way as to be able to remove the outer skin from the tongue after it is cooked, which means cutting the tongue into chunks that are larger and can be skinned after cooking without too much fiddling.
The consistency of cooked barbacoa is not that of shredded meat, it has a very fine texture, fine enough to be eaten with a spoon. The secret therefore is long slow cooking, but also not to use a lot of liquid. I have read lots or recipes with the sentence "after cooking discard the liquid" in them. This makes me cringe. Neither is is necessary to add a lot of competing flavors. Some garlic and some salt are all that is necessary. A bay leaf would be fine but anything beyond this, chile for example, is overkill and detracting for the softness and beefyness of the dish.
Cooked on low overnight the following morning all that remains to do is shred the cooked meat very finely, reduce the cooking liquid and add it to the meat until you have a very fine smooth texture, almost a loose paste.
It's a wonderful Sunday morning brunch and needs nothing more than some chopped coriander and onion and a nice tomatillo salsa to make some excellent tacos.
The first time I had barbacoa I was very surprised by it. The name suggested something completely different to me; something grilled, and the texture was new. There is something elemental about it's simple beef taste. If there was a periodic table of cow, this would definitely be an entry on it.