Is there a way to track which fans are sharing my Facebook Page?

No.

No, not at the individual user level, and Facebook probably isn't too keen on enabling marketers to do this anytime soon. They received quite a bit of negative PR from the Wall Street Journal Expose on how Rapleaf was tracking individual Facebook users without the user realizing it. Instead, IF Facebook enables this in the far-future, it will probably be something driven by consumers, where consumers have the option of letting the brand know they're some kind of "superfan" and sharing the page. (If history is any indication, it will probably also be turned on by default, and many Facebook users won't realize it.) However, you can see how many people are joining your fan page because a friend invited them - see:

It is not possible to track views as such. There are a number of tools (including my http://www.brandbook.sm) that track the "most prolific fans" - ie those that leave more comments or likes more often. The question is my opinion is another. Lets say you are a multi million business and have a multimillion fan page, and let's assume you could track which fans interact with you page. What do you do with that info in order to monetize it, since you do not have the demografic data (except at times gender) of your fans? As much as the objective of an add is not to be as funny as possible but to sell, the objective of fan pages is to reinforce the brand, not to maximize the number of likes or of comments.