I think it is important to decide beforehand what kind of work experience you are looking to have?
This may be old fashioned of me but I would strongly advise against an email as a first introduction. Emails can be overlooked, deleted, received as spam, etc. An actual letter, typed and well written, is more likely to be paid attention to because it shows you put some effort in to it rather than "flicking off some emails..". Once you establish contact, then let the restaurant decide how the communication will continue.
Yes, there are thousands of restaurants, but there are not so many great ones. Before contacting them you should be doing some research to find out which ones are best, why they are important and what makes them unique. A simple google search for top NYC restaurants is enough to get you started. Do some reading about them so you can decide what kind of experience you are trying to get and which restaurant can best provide it. Once you have narrowed it down, then send a letter written specifically to those restaurants.