You are talking about John's Pizza.... by far it is our favorite place to eat. The one you are referring to is on 44th street between 8th avenue and Broadway. The original one is on Bleecker Street in the Village.... I prefer to eat at the Bleecker st. location.

As for the dirty water dogs I have seen where a lot of these vendors store their food and it isnt pretty.

There is a 2nd Carmines up on 91st street and Broadway that isnt the tourist trap that the Times square location is. The prices are the same with half the wait.