That's not going to come out like you think it will. It's a box that was built on-site, and is firmly fixed to the studs with finishing nails that are hidden beneath those layers of paint. The only way you're getting that out is to remove the mirrored door and the trim that surrounds the opening, and then have at the individual pieces by prying them away from the studs with a couple of Wonderbars. This will be grunt work, and you may need to get a hacksaw blade inserted between the wood pieces and the studs to cut through some of the nails once you make some progress prying.

I dealt with a medicine cabinet almost exactly like that in my daughter's house recently, and I finally decided the best thing to do was to replace the wood trim with new pieces, construct a new, mirrored door with Euro hinges, and leave the box alone. All I did to the interior was sand it and repaint it.

Thanks. That makes sense. I saw this tutorial on HGTV that creates a new medicine cabinet mirror for a recessed wood box.

https://www.hgtv.com/how-to/home-improvement/how-to-diy-recessed-medicine-cabinet-modern-mirror