Closing piano music courtesy of Harpeth Presbyterian Church, used with permission.
The way Wikipedia tells the story, this took place at the Earl’s home. I suggest it was more likely at a 
 “gentlemen’s” club in the afternoon.
Now here is where the story becomes murky. Knowing the sensibilities of the time. The slab of bread was on a plate (hopefully small) and accompanied by a napkin knife and fork! 
It’s not clear as to what happened next, did the Earl of Sandwich fold over the bread and pick it up?
It is clear that other players said “I’ll have what Sandwich is having!” And the name stuck. 
We would know it as an “open-face sandwich”!