Are you talking about wanting to try the sauce or the chicken over rice?

Halal is, like you mentioned, basically preparation rules. Like Kosher (very much, in fact) it is not a dish or even a style itself. So lots of things can be had at Halal carts. New York just happens to have a large enough population demanding Halal prepared foods that lots of carts sprung up. It is a good thing.

Several dishes could fit what you are describing, but I am betting that you most likely were looking at Chicken Shawarma. The chicken was probably either seasoned and grilled, or done on a vertical split, right? The rice was probably just a pilaf, or even just rice. The sauce most likely was some version of Tajik/Tzajik/TzaTziki sauce - a yogurt based sauce, almost always with garlic, and usually dill or cucumber or mint or a whole bunch of different things. Personally I hate the stuff, but most people love it. It is found throughout the region, and there are hundreds of different recipes - the sauce in some cases is what makes one places food special, not the meat.

Not sure where you are from, but try and find a sandwich or pizza place that sells shawarma. It will be slightly different seasoning than you are used to, but in the end it is just one heck of an amazing chicken sandwich when done well.