There's no way Chick-fil-A does it. I got a few bites into the Popeye's sandwich and then realized they weird, off putting flavor was a pickle brine. I order all chicken sandwiches plain, since I have a really sensitive tongue and can't stand the taste of pickles. The only time there was a hint of pickle flavor in a Chick-fil-A sandwich is if the person who bagged the sandwich had pickle juice on their gloves when they grabbed the patty.

I think they just salt brine their chicken and the common misconception that they pickle brine comes from the fact that pickles are a vile vegetable whose juice soaks deep into whatever they touch and most people order them with pickles.