How should you write a letter? You shouldn't.
A graduate degree may help you get a job. It may help you get a better job, which may mean a higher paying job. But the time to negotiate starting pay was before you started. Having accepted what they offered, they have no reason to give you a raise until you earn one.
Put that additional knowledge to use, do things better/faster/more robustly than other employees, and show the company you're worth more than what they're giving you now. There is no entitlement here.
You are free to put the dice back in the box and start another job hunt. But you don't exactly have a good story to tell when they ask why you want to leave so quickly.
Personal opinion: This sounds like a combination of not understanding the system and "buyer's remorse". I'd suggest you stop worrying about what anyone else is being paid; it really is almost completely irrelevant right now. Instead put your attention on getting a high performance rating so they can justify giving you a raise when the next review cycle occurs.