If you cut a sandwich in half, do you have two sandwiches or one sandwich, made up of two halves?
 
Discuss.

One sandwich, 2 halves.  You don't cut an apple in half and have two apples.

" One sandwich, 2 halves.  You don't cut an apple in half and have two apples. "

The only part where that becomes questionable is when it comes to party platters. Those are always chopped up into many pieces, thus making it hard to discern how many pieces made up the original sandwich, so each piece is typically called a sandwich. 
At my workplace, there are sandwiches made up of pita bread, no crust, so it's even harder to tell how much makes up a whole sandwich. It's usually four pieces per sandwich, but they're so large that each individual bit is considered a sandwich. 
 
So I'd say it all depends on the amount of cuts, the size of the cuts, and the type of sandwich that is being made.

If someone cut a sandwich in half, and gave me one of the pieces, they would have given me half of a sandwich, not a whole sandwich. So one sandwich, two halves

that is a good question. Realistically you have a sandwich in two, technically it may be two sandwiches.

It is only one sandwich until you give one of the halves to someone else in which case it would be two smaller sandwiches though they individual sandwiches can still be referred to as a half. I.E. "That's my half" or some such.