What surgeries or procedures can be done for very deep set/sunken-looking eyes?

This surgical option is really not advisable, since it can cause double vision and many other problems. But it is the treatment for enophthalmos ( deep set eyes). The best treatment for you , I believe would be fillers around the eye. That is much safer and the results are very comparable. 

Most expert experienced artistic plastic surgeons have a
variety of procedures, both non-surgical and surgical, at their disposal to
produce excellent cosmetic results in the appropriately selected patient: Botox
or Dysport, temporary hyaluronic acid fillers like Restylane, long-lasting
micro-fat grafts, blepharoplasties, or combinations of procedures. Each has
benefits and drawbacks and depends on desired outcome and individual anatomy.

Following the advice of anyone who would presume
to tell you what to do without seeing photos and without taking a full medical
history, examining you, feeling and assessing your tissue tone, discussing your
desired outcome and fully informing you about the pros and cons of each option
would not be in your best interest. Find a plastic surgeon that you are
comfortable with and one that you trust and listen to his or her advice. The
surgeon should be certified by the American Board of Plastic Surgery and
ideally a member of the American Society for Aesthetic Plastic Surgery (ASAPS).
You should discuss your concerns with that surgeon in person.

Robert Singer, MD FACS

La Jolla, California

In some cases the improvement of the appearance of eye hollows or "sunken" eyes can be accomplished with dermal fillers or fat transfer. Thanks for your question. Best wishes.