There is some degree of controversy surrounding the invention of the horseshoe sandwich.  It is undisputed that it was created in the first half of the 20th century in Springfield, Illinois.  The most widely accepted version credits chef Joe Schweska at the Leland Hotel in Springfield in the late 1920′s.  The story goes that the original horseshoe was so named primarily because it was served with a boned slice of ham which was horseshoe shaped.  It had the same ingredients that we still enjoy today, bread, meat, sliced potatoes and of course cheese sauce. 

Here is a  cool now and then picture of the Leland Hotel.

http://www.springfieldrewind.com/leland-hotel-1950

And here is another discussion of the history over at What’s Cooking America

http://whatscookingamerica.net/History/Sandwiches/HorseshoeSandwich.htm