Am heading to New York in May
New York has around 5000 restaurants so your query is hard to answer. I will say that when I was there last month I headed for Ipanema in the Brazilian block of W 46th St between 5th and 6th because I wanted a good feijoada completa, which I can't really find in Chicago. The ambience was fun as I heard only Portugese all around me and everybody else's lunch looked as good as mine. Not sure what would be an "amazing experience for you" but I definitely hit Ipanema every time I am in the city and am sufficiently amazed.
Really been enjoying these lately:
Blue Ribbon Izakaya
Had an excellent dinner at Kin Shop last night (same owner as The Marrow & Perilla).
It's hard to find a completely hidden gem these days and I am not sure you can really fault a great restaurant for being discovered by the media... Especially when the media reads food boards like these. Underrated, yes. Completely hidden w/ no presence on Yelp/Chow/etc, not really.
http://newyork.seriouseats.com/ - their neighborhood guides especially have some lesser known gems.