I love sandwiches, they are probably my favorite food. I have a special affinity for regional specialties. Leaving burgers out of the conversation, my top 5:
Katz Pastrami - rye bread, deli mustard, and a huge amount of that amazing cured and smoked meat, it melts in your mouth. The corned beef is also excellent.
Maine lobster rolls - I've been to Maine twice and if I had to leave NC it would be near the top of my destinations; many lobster rolls have been consumed, but Even Tide stands out for the non-traditional style.
NC BBQ Sandwiches - Wilbers in the East, Red Bridges in the West, Stamey's and Little Richards in the middle, I don't choose a side in the East v. West, they're both great. If you're in the Raleigh area, head to Prime in Knighdale.
Po Boys - I don't really like New Orleans but seafood was made to be fried and even better with some dope sauce and good bread.
Fried Chicken - I love the Nashville style, Buffalo style, or the ways Ashley Christensen does it at Beasleys (either hot with the Carolina Reaper oil or regular).
ETA: all go exceptionally well with cold beer.