A framing shop knows how to do one thing. Frame art. They do not genrelaly know anything about building cabinets. Picture frames do not involve a lot of hinges.

Our medicine cabinets don't rust because they are simple wooden frames built into the wall, In the picture below, the middle mirror is fixed and the other two mirrors are the doors for the two cabinets. The doors are cabinet grade plywood with Blum hinges, and the each mirror sits on a nearly invisible piece of stainless steel trim and is held in place with mirror adhesive. Inside, the sides of the boxes are drilled so that adjustable plexiglas shelves can be set on pegs with whatever spacing is desired. This installation is 15 years old and is like new still. .

Pegasus makes some that are not quite, but almost, round. https://www.ebay.com/itm/Pegasus-511335-Oval-Gold-Frame-Deco-Mirror-Medicine-Cabinet-29H-x24W-x5-5D-NIB/332734951857?hash=item4d788c71b1:g:pN0AAOSwtuBa4-Uq 29 x 24 is getting closer! FOUND one on eBay! https://www.ebay.com/itm/Signature-Hardware-Ellipse-Stainless-Steel-Medicine-Cabinet-with-Round-Mirror/163145668458?hash=item25fc3d576a:m:mdq0mOfSGyx8lWvADxfgGbA&var=462310658746&_sacat=0&_nkw=round+medicine+cabinet&_from=R40&rt=nc&_trksid=m570.l1313 See what I mean about tiny? This is the only kind I have seen. The roundness makes opening the hinge a problem if they get bigger.

Any med cab that is more than 14" wide requires that a carpenter make the studs in the wall "special" like your original house. They simply design the wall supports (studs and headers) to fit around the medicine cabinet. If you look at the triple cabs, you can see that there is no 2" stud between the cabinets. Any wooden cabinet will have a thicker back wall than any metal cabinet, reducing the interior space or forcing the cabinet to protrude further out of the wall to get the same depth. You can chose a cabinet that protrudes out a bit in order to get a deeper, more versatile cabinet, if you do not mind the edge sticking out beyond the wall. I would much rather have that extra depth be usable, not the thickness of a wooden back.