I am not a woodworker. But wouldn’t it make sense to reverse the process. Have your cabinet maker cut a blank slab to the size of the full cabinet. Install the hinges on that. And then, have him affix a mirror to the slab and frame that with your frame of choice. Don’t put the hinges on the frame at all, but on a piece of wood that the hinges can be attached to and will hold the weight of the mirror and wooden frame.

Ptreckel,
Thanks for your comments! I thought that your idea would work, and it would have made more sense for me to have waited to make the custom frame. Alas, the custom frames (we have 2) already made and we have to figure it out around it. The thickness of the frames make it difficult to find hinges. The other possibility is to take that blank slab as you say with mirror (that is smaller than my 24x36” mirror to fit inside recess of frame) and add another piece of wood behind it so that any kind of hinge piano or European style) would never touch the frame. For some reason, our cabinet maker was so skeptical of the idea and he said he would need to cut notches in our frame, that he backed out of making these cabinets for us in the end. Now we either have to find another cabinet maker, figure it out ourselves, or scrap the idea and just put up mirrored frames without medicine cabinet (such a shame of labor, with tiles etc built around medicine cabinets) with space behind these frames but unable to use it!

And we were supposed to move in before the holidays, but it doesn’t seem possible now.

If anyone else has any other ideas, we would be most appreciative. Thanks!

I think ptreckel's idea is the one to work with. You’re not going to be very successful in using those frames, custom or not, because the rabbet, the cut channel where you intend to put both mirror and wood, is not deep enough and that frame won’t be sturdy enough to hold up.