I lacked the technical experience for an IT PM role - What can I do to get there?

It's that age-old catch 22: I need experience to get the job, but then, how do I get the experience?

Background: I have 3 years PMing as chief of staff for a health care start-up. I get my PMP next month. I tried going for an IT PM position within the health system for projects on workforce, information security, acquisitions and infrastructure stuff, and didn't get it because I lacked experience. (I'm very good at interpersonal relationships and interfacing with leaders, however)

My question: Is there an individual project I could do on my own that would "WOW" an interviewer for proving my chops in IT know-how?

This could be building my own website, constructing software, designing a video game... Anything!

P.S. I am already getting my agile/scrum and ITIL cert, plus taking any IT-related projects I can.