The best chicken sandwhich can be found at Whitewater Snacks in the Grand Californian. Just use the hotel exit from DCA and turn left at the end of the hall. It is large and a good deal for the price compared to other places in DLR. I've had the burgers at Hungry Bear and they are mediocre at best. The chicken sandwhich at Whitewater Snacks comes with guacamole and roasted peppers but you can have them off if you don't like them. For me, that makes the sandwhich. The meat is a big piece of white chicken. It is also far superior to the chicken sandwhich at Taste Pilots Grill.
Here is a pic. You can see they cut it in half for us. DH and I split it and I was full. This is now a must-have every trip.
As an added bonus WS is also a nice quiet place you take a break from parks in. It is very uncrowded and relaxing.