I know this is an old thread, but I faced the same issue: surface mounted medicine cabinet 4 7/8" deep. I found a Kohler Tone 3-light vanity fixture Model #31770-SCO3- BNL (brushed nickel one) that was perfect. It extends 10 3/4 inch out, but it's only 6 inches high. It was beautiful, but after ordering from Lowe's, it went from delivery 6/12 to delayed to out of stock. Damnitall. Whoever reads this...it will be available in October, 2023. I talked to a Lamps Plus clerk who said the entire bulb must fully extend out past the medicine cabinet to avoid all shadow. This Kohler model does that. I had to totally switch gears to a fixture that pointed up. I hope it will work to provide enough light. I'm not as thrilled about it, but you have to be practical to get the bathroom remodeled in time. I hope this is helpful. Good luck to all who face this issue!

No use in reinventing the wheel. (See Link.) If you have plaster walls with wood lathe, all bets are off. As soon as you start sawing, the old plaster starts to crack and fall apart. Here is a link that might be useful: Installing a Medicine Cabinet

If you wanted a deeper cabinet, you could recess and have some sticking out ;) too. A third option! I've had recessed in old bathroom and it was fine. I now have a new surface mounted one that just got put up this friday. I have an exterior wall in cold Iowa. It was made for me by my fab carpenter to go with an old Robern medicine cabinet mirror door I've had for 20 years (got it for free in an alley outside a design shop-she said "take it"). With the recessed and want to change the set in the future, you'd have to do major wall repair or find a cabinet that fits. But wouldn't that be unlikely? If I didn't like the protrusion, then recessed is what I'd do.

I looked at the cabinet spec and the attachment point are on the front end of the cabinet box, so you would have to make your own holes in the back to attach the cabinet and be sure that the material will hold the weight of the box + mirrored door. Also the mirrored door is 8" larger overall than the box, so how do you plan on disguising that? I would sell the mirror and buy one designed for the way you want to use it. Any hacks will probably end up costing way more than just buying a new medicine cabinet.