@Gucek - With some careful woodworking, you should be okay. I think the main issue is going to be how do you secure both the glass and the monitor so that they don’t fall out when you open the glass/frame component. You don’t want the glass to fall out and you don’t want the monitor to fall out.
My very first mirror had a 3/4" gap between the mirror and the monitor and it was noticeable. You will want to find some way to secure the glass so that it’s held securely but presses flush up against the monitor when closed.
Take your time and plan out everything. Good luck.