Ok, so my suggestion isn't really west Texas but just up the road on I-25 in Hatch NM (no more than 80 miles from El Paso)...
Sparky's is by far the best green chile cheeseburger I have ever eaten ( and I have spent a 60 yr lifetime searching). Get the one topped with world famous Hatch green chile and bbq'd pulled pork while sippin on an El Rojo and listening to local blues bands.
http://sparkysburgers.com/