If I was trying to get into a PhD program at Princeton, I would find a way to get connected with a professor in the EE department. A graduate student is more like an employee than a student. You're helping a professor with their research, teaching, and at the same time working towards a PhD. Treat it is if you're looking for a job. Don't expect your folder to rise out of the pile of folders of applicants - you need to find a way to get some push behind you. Talk to your professors and other contacts and see if one can connect you to a professor in Princeton's EE department (as well as professors in the EE departments at other universities).
If you can't get a connection, write to a professor whose research is aligned with yours, and then follow-up with a call. You'll have a lower batting average doing this than if you get a connection, but you'll have a much higher batting average than if you simply apply and hope they choose you.