JK - Weinberg's does serve tongue! It's an option on their sandwich menu. Not a New Yorker but liked their sandwiches. Went for breakfast once and the waitress wanted to make sure that my husband knew that lox was a lot saltier than Nova smoked salmon.
Have also liked the vibe of Big City Bagels (which is in North Raleigh and still far from OP for a regular hangout) and there are a lot of NY people that go there. But just noticed on Yelp that it's under new ownership and things may have changed. We haven't been in a while so I guess I can't comment on that anymore.
Your location is not bad to get to Raleigh, Durham and Cary for your food fixes, which you will probably need to do for what you are asking for. On the bright side, all the good ethnic food is nearby!
In case you haven't seen this thread from before, it lists most (if not all) of the locally owned Cary restaurants. Looks like adlnc might be able to give you some good Morrisville tips too.
//www.city-data.com/forum/ralei...staurants.html
Check out Poppyseed Market in North Raleigh - again a little further away from you but a good array of subs and deli sandwiches and there is some chicken parm on the menu as well.
Gravy, in downtown Raleigh, consistently gets pretty good reviews for food. They also have chicken parm and gnocchi on the menu although I will also attest that Melina's gnocchi, as noted by Sal_M, is quite good!
For Latin - Durham will be a good place to find a lot of that, as well as the Capitol Blvd area in Raleigh. Balcazar Bakery is a Columbian bakery, there are some other Salvadorean eats over in that area as well, as well as a bunch of Mexican places. Don't know about arepas though outside of Guasaca (and buku, which tries hard but not in the same league). Downtown Raleigh also has Oakwood Cafe, which has Argentinian food. I also really like Macchu Picchu for Peruvian food over on Old Wake Forest Rd as well as my favorite Peruvian Rotisserie chicken from Mami Nora's on (regular) Wake Forest Rd.
You may also like this book as a baseline:
Food Lovers' Guide to Raleigh, Durham & Chapel Hill: The Best Restaurants, Markets & Local Culinary Offerings (Food Lovers' Series): Johanna Kramer: 9780762779765: Amazon.com: Books
IndyWeek also puts out an annual food issue that is pretty comprehensive. And the food blogging scene goes through fits and spurts but there are some good ones out there that stay updated. And ones that stay on top of the food truck scene. Lots to explore in all different food genres. I predict your weekends will be busy travelling to Durham and Raleigh to try things out...and of course, it goes without saying that if you find your go-tos, you should post back and share!