Cattlemans Steak House
I am struggling to recommend whether I should recommend you cut across El Paso on Trans-Mountain Highway or not. If you were going both ways, I would say you should take both routes (I-10 and Trans-Mountain), but as you are only going on one direction, you should probably stay on I-10 and see downtown El Paso.
If you look to the right as you are entering El Paso, you will see a big hill with a cross on the top. Mexican Religious Pilgrims climb this as a Holy Adventure, and are often robbed by banditos.
Also as you are coming across through Las Cruces, you should stop for sightseeing, and and perhaps dinner in Old Mesilla. Get some of the dishes with the Mole Sauce (a traditional favorite that includes chocolate (coacoa) in the reciepe).
Historic Mesilla
If you don't have time to go to Mesilla, but are hungry, get off in Las Cruces and eat lunch at Dick's. They have GREAT green chili cheese burgers...
Yelp - Dick's Cafe, Las Cruces
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