I work at 51st and 6th and am a really big chicken and rice fan, so I've sampled quite a few of the carts. Note that I only eat chicken when it comes to street meat, because I don't eat mammals. So YMMV if you're a lamb or beef fan.
My personal favorite carts are Kwik Meal at 45th and 6th, the Biryani Cart at 46th and 6th and the Halal Guys at 53rd and 6th. Each has its own high points.
At Kwik Meal, the rice is a bit more flavorful, the salad includes some Italian dressing, the white sauce is more yogurty and they have this special green sauce that's really unique and delicious. But their portions aren't huge, so sometimes I'm still hungry afterwards (unless I shell out for extra meat).
At the Biryani Cart, the kati rolls are absolutely delicious (if a bit greasy). And I'm a big fan of the chicken tikka (or the combo kebab, which includes both chicken tikka and "chapli," which is kind of like a cross between falafel and a chicken-burger). They've raised their prices a couple times since midtown lunch first popularized them, and then they won the People's Choice Vendy, so it's not as good a value as when I first started going there. And sometimes when they're really busy the quality isn't 100%. But the food is truly delicious.
53rd and 6th is a little less unique than Kwik Meal or the Biryani Cart. But it's the best value by far in terms of food per dollar, and I find it to be less greasy than most other street carts.
There are two important things you have to know about 53rd and 6th, though:
First, the "famous" cart is on the southwest corner only at night (they set up around 7 or 7:30pm and stay until 5am--or so I'm told). During the day, somebody else is there that's truly inferior. I've tried quite a few of the standard chicken-and-rice plates you can get at various carts in the surrounding blocks at lunchtime, and none of them (with the exception of the next paragraph) is as good as the nighttime 53rd/6th SW corner guys. Some are too greasy, others just don't have good flavor. The SW corner of 53rd and 6th at lunchtime is particularly bad. At one point, the nighttime guys even had a sign saying they "weren't responsible" for the food served during the day at that spot. My guess is that the daytime guys got pissed, though, because that sign disappeared after just a couple days.
Second, there is an all-day cart on the southeast corner of 53rd and 6th. I'm not sure how late they stay, but it's definitely there at lunch and dinner time. They have more or less the same thing as the guys on the southwest corner, and according to Midtown Lunch the owners are cousins. They have the same shirts, the same umbrellas, the same bags, the same containers and (most importantly) the same white sauce. Sometimes when they are not busy, their chicken can be a little more dried out than the SW corner guys, simply because the SW corner guys have such insane turnover. They also sometimes have tomatoes in their salad (the SW guys are usually lettuce-only). And unlike the SW corner guys, they offer a couple other options, like falafel and gyros, although I've only ever gotten the chicken and rice. But in every other way they are the same thing. And during lunch, the same plate that costs $6 at night costs only $5, which IMO is one of the best value lunches you can find in midtown.