two usually hinged boards designed for hanging from the shoulders with one board before and one behind and used especially for advertising or picketing
Check the sandwich board out front for daily food specials.
You generally see sandwich boards on restaurant patios. They'll typically take the form of a blackboard and list daily food specials.
So, in this case, sold off the sandwich board just means that you've ordered food from a selection that's listed on one.
I have no problem understanding sold off the sandwich board. The missing preposition is assumed, as in cases of other sentences with elided words. Perhaps in strictly formal writing it would be an issue, but here it fits with the stylistic sense of the rest of the article.
People can buy items off the lawn in yard sales, and they can order off the menu in restaurants. Restaurant owners who sell off a sandwich board do the same sort of thing (grammatically).
The missing preposition aside, which I don't think is a real problem, the only issue is that, at first glance, it can be taken in an ambiguous sense as there is another expression along the lines of I sold off my valuables. Ironically, in this case, "off" is added to the phrase when it would be clearer without it (simply I sold my valuables), but people say it anyway and it's understood.
So, in this case, there is possible confusion between one type of informal phrase and another. Interpreting it the "wrong" way would imply that the restaurant had a menu board that was, itself, sold. But I think that at worst it just slows down comprehension a little bit.
My own problem with the sentence is understanding why you can't share food that is sold off a sandwich board.
Looking at the source, it's talking about about "mixing and matching" food. But that, still, has nothing to do with sharing with other people. (You can share "mash-up" food and you can share sandwich-board food.).
Taking the source material into account, I think the sentence would have worked better if expressed as:
Our reactions to these foods are often instructive about our own prejudices on how to eat. Sharing your own mash-up food is so much more of a joy than ordering from a restaurant.
(I do have to say that the whole article is written in an unusually complicated manner.)