Certainly not Chick-Fil-A:
Sandwich, in its basic form, slices of meat, cheese, or other food placed between two slices of bread. Although this mode of consumption must be as old as meat and bread, the name was adopted only in the 18th century for John Montagu, 4th Earl of Sandwich, who had sliced meat and bread brought to him at the gaming table so that he could continue to play as he ate. His title lent the preparation cachet, and soon it was fashionable to serve sandwiches on the European continent, and the word was incorporated into the French language.
As a curiosity, the Bible cites something similar to a lamb sandwich in Numbers 9:11:
The fourteenth day of the second month at even they shall keep it, and eat it [lamb] with unleavened bread and bitter herbs.
Using Google Books, the
first mention I can find of a "chicken sandwich" is dated 1827:
Google was wrong, the earliest I can find is dated 1841.
[...] and who, having refreshed her guest firstly with a glass of soda water fresh from the ice basket, and finally with a most unexceptionable cup of coffee and chicken sandwich, considerately left her to the repose which a delicious couh under a waving punkah, steadily pulled during the night, irresistibly invited.
See also the relative NGram.