Chicken Giardiniera, AKA Italian Picnic Chicken Sandwich, or to Calabrese in Melrose Park, simply "Picnic Chicken."
This is a sleeper and was brought to mind by the Popeyes furore. Traditionally it's fresh fried pounded chicken breasts with generous parmesan / romano and black pepper in the breading on crusty Italian or squishy white bread with giardiniera as the only condiment / inclusion.* Ideally the sandwich is served cold, or rather, chilled, and then served room temperature at a picnic.
A leftover is making the sandwich from a bucket of pulled Brown's chicken legs and breasts (on Turano with giard, or if nonna was making focacch', that route).
This is more a distinctive home cooking thing than a restaurant thing but I've seen it in the wild at Franco's (31st) and the now-closed Andorka's (Halsted).
*If you have a tomato, it's not going directly on there. Cut it up with basil and vinegar as a side dish and use the sandwich to soak up the juice.