This is chicken in bread that has not been turned into a breaded cutlet. Soy-and-ginger marinated karaage, teriyaki-glazed thigh, the vinegar-and-tartar chicken nanban, grilled yakitori, sweet-spicy Nagoya wings: the chicken is fried or grilled and sauced rather than crumbed, which is what separates it from the katsu sando. The defining fact is a juicy, sauced piece of chicken carrying the sandwich on its own flavour.
The craft is the cook and the sauce balance. Karaage is marinated and given a light starch coat so it fries crisp but stays juicy; teriyaki is glazed so it is glossy and savoury-sweet without going sticky-heavy; chicken nanban is dipped in its tart sauce then topped with tartar so the richness is cut. Mayonnaise and shredded cabbage often go in to add cool and crunch against the warm meat, and the bread stays soft and plain.
The variations are the preparation: karaage, teriyaki chicken, chicken nanban, yakitori, the Nagoya tebasaki wings. Each is a different chicken cooking method given soft bread, and each deserves its own article rather than being crowded in here.