Achieving more deep-set eyes
As we age, we lose fat volume in our orbits and our eyeballs naturally sink in. The same happens if FOR ANY REASON the volume of the eye socket is made larger (such as a fracture of the floor of the orbit, removal of part of the side wall of the socket (as is sometimes done in hyperthyroidism where the periorbital fat increases in volume due to inflammation).
It is really NOT complicated to cause the eyeballs to fall more into the socket. The question is how much floor or side wall widening needs to be done ? and Is it worth it? Uniformly, the long term result will be poor. It will prematurely age you.
Personally, I do not feel such an operation is an ethical operation in your case NOR would I want to be associated with such a result.
The eyeballs cannot be made more deep-set into the orbit of the face. Fat pads when present on the lower lids can make the area look puffy, these can be removed to create the illusion of a more inset eyelid. However, the puffiness has to be present first prior to removing any of that lower orbital fat.
Have your thyroid checked
Bulging eyes or exophthalmos can be caused by metabolic/endocrine conditions such as hyperthyroidism. Start with a visit to your internist for some simple blood tests to rule out these conditions. Then, you might want to see a board certified plastic surgeon for options to correct your eye appearance. Good luck!
Trying to make the eyes deeper set in the orbital cavity is usually not done for patients unless they have sever exopthalmos. When this is performed, usually out fracturing is performed to make the cavity larger to accomodate the eye.
Wanting more deep set eyes is an unusual concern
This is an unusual concern that you are describing. It is the nature of the world we live in that you may find someone who will try to do whatever you ask. However, you need to be carefully assessed to determine if what you are considering makes sense. Orbital decompression with fat and bone remove is used to set the eyes back into the orbit when thyroid eye disease pushes the eyes forward. However, it is much less common to use these surgeries to perform "comsetic' orbital decompressions for the purpose of moving the eyes back.
My best advise is to seek several opinions for expert ophthalmic plastic surgeons. These are board certified ophthalmologists with specialized training in eyelid and orbital plastic surgery.
The American Society for Ophthalmic Plastic and Reconstructive Surgery maintains a directory that will help you located a quailified doctor near where you live: www.ASOPRS.org.
Deeper eye sockets: It can be done, but it should NOT be done.
Orbital decompression is the name of the procedure done to achieve what you're describing. But it's an operation reserved for significant eye problems from diseases such as Graves' ophthalmopathy. A frequent side-effect of such decompression surgery is diplopia--double vision--which has to be subsequently corrected by additional surgery on the eye muscles.
From a cosmetic standpoint, the risks don't justify the procedure.
All the best,
Not possible to deepen your eyes
What you are describing can be done for certain extreme conditions like hyperthyroidism but for purely cosmetic purposes, this would not be safe to try.