Install a recessed medicine cabinet with stud problems

Measure from your closet stud on either side to the other. Take that number and subtract it from your drop-in cabinet. No take that difference and split it in half; the sum will be how many inches you'll need to shim if you have the cabinet square in the first place. If you have something smaller then a 2×4. Left over (say the distance is 5 inches) obviously a single stud shim will not work you have a few different options: rip a 2×4 length wise to the desired measurement. Or use 1× 2. Either o which are safe since it's only a medicine cabinet. Just get the thing square and plum. Use a level and double check your measurements before you cut. It's a finished wall not the most fun to repair if you like a seamless wall. If the depth of the wall isn't as deep as the cabinet your reveal may be funny looking.

I'm hearing that you don't want to cut out drywall, install bridge framing between studs, put your previously removed drywall back in, spackle, prime & re-paint. Well, this Medicine Cabinet better be 24" wide or less.

If so, then you can simply screw through the drywall into your new framing sticks. Do this from both sides of the wall for optimal strength & support. Or, turn the 2x4's flat against the bath wall & install 2 rows of screws to secure them.

You'll want your top & bottom pieces longer so you can toenail the side up & down on the left side. The right side will be a little funky, but that's not entirely your fault. Finally, you can frame the Medicine cabinet like a picture to hide all of your short-cutting wall damage.