Have you looked at the audio libraries that are available?
https://www.pjrc.com/teensy/td_libs_Audio.html
That content seems old but maybe it'll help or at least provide some direction. Personally I've only done predetermined audio file playback with a button press, but constant recording sounds like something that would be possible. True constant rewriting on the EEPROM is from what I understand not the best idea because of how quick you'd wear it out. A fast sd card might be the ticket there. Good luck, it sounds like a fun project!