Here is what one of my friends experienced.
Buyer Beware, though! You CANNOT purchase the Karkand expansion through Origin and give it as a gift IF YOU ALREADY OWN IT. You have, basically, just purchased it for nothing. There is NO Warning about this on the Origin purchase page. I had to search the EA help page to find anything along the lines of "Oh, by the way, if you already have the Karkand expansion (through the Limited Edition), you can still purchase it on Origin (though nothing happens), so be careful." I received an email with my purchase order number, and nothing else.
I then emailed EA/Origin and explained my dilemma (and explained their shortsightedness for not, at the very least, putting a WARNING in the Origin Store about this). To their credit, they did email me back fairly quickly (I had already gone to sleep at the time), giving me an actual code and the steps to take to load it up. I've forwarded that to my brother, so he should be able to play today.
If you purchase the expansion from elsewhere, Amazon for example (that's where I got the Limited Edition in the first place), you will probably get that code and you can give it as a gift. Do NOT attempt this through the Origin Store to give it to someone else, or suffer the same fate as I. Dump dump DUMMMMMMMM. ... seriously.
So, if you already own a game, you cannot gift it...but if you don't the game you want to gift, that is a good question...