I have a REAL aversion to most cold cuts but especially "roasted" turkey -- slimy and weirdly pink -- so I was delighted to remember (and then eat at) Mr. Henry's on Cap Hill, 6th and Penn. SE, across from Citibank.
Full disclosure: I was once a waitress there and had a great experience (many gay customers, meaning many great tips, I think because so many were waiters themselves), and was favorably impressed by the kitchen: no weird smells. Great patio with umbrellas on 6th Street. Good prices. (worked there 10 years ago so feel like the ethics clause no longer applies to me).
ANYWAY: great lemonade with actual lemons. Great potato chip/fries. Cut from real potatoes (boiled first to speed frying). And they roast turkey breasts every day for the turkey sandwiches and turkey clubs. Nearly as good as day after Tgiving samiches. Ask for a little extra mayo on tyhe side because the turkey -- like the ones you make at home, admit it -- can be on the dry side. The turkey is a little expensive at $6.95 per sandwich but 1) it beats waiting till thanksgiving and roasting your own 2) quality food prep should be rewarded.
I simply refuse to eat another slimy piece of turkey again and because I live around the corner i do not have to (and neither do you...E Market metro). Mr Henry's also does good burgers etc and on Friday nights has a smoke free jazz club upstairs. Not sure if connected to the other Mr. Henry's around town (Gtown and Adams Morgan? Not even sure if they still exist...)
As for real chicken sandwiches: I actually HATE wraps but there is one I eat with gusto -- the roasted chicken wrap at Canales Deli in Eastern Market (near C Street entrance). Real chicken, roasted and pulled and packed in a good flour tortilla and served with this weird green hot sauce of Cuban origin, I think.
Cheap and delicious and above all not sliced off a chunked and formed roll of "meat."