This is what I found in NYC for anyone interested. I haven't searched for chicken yet, just beef and some pork:
Petrosino on Houston Street has both organic steak and pork loin:
Of course for a burger there is New York Burger Company which I'm looking forward to trying (I've been to Better Burger - it was just OK in my opinion).
This is all I've found in Manhattan. I know there is more - but they don't make it easy to find.
As mentioned by emiloo there is Applewood in Brooklyn which I'm excited to try and also
Bluestone Bar and Grill
Would love to know anyone's experiences with these places...thanks in advance!