Off hand, I know of one very famous one-eyed pilot. His name was Wiley Post. He became a pilot after losing an eye in an oil well accident. He went on to set several world records, including two flights around the world (one with a navigator and one solo) in his Lockheed Vega, the Winnie Mae. That plane is in the Smithsonian Air & Space Museum (along with Amelia Aerhart's Vega).

In addition to setting speed and distance records, Wiley Post pioneered high altitude flight. He invented a pressurized suit, the forerunner to the space suit, and appears to be the first pilot to experience the jet stream. Post was killed along with humorist Will Rogers in a crash while flying in Alaska back in 1935.

I'm not certain how Wiley Post handled depth perception with only one eye, but he obviously did. I've read that he developed a trick of turning his head from side to side while landing to help him judge altitude. Apparently, he also used some form of mental calculations based on the known heights of surrounding buildings. However he did it, Wiley Post was a remarkable pilot and innovator.

As for other one eyed pilots, there appears to be quite a few of them. According to this source:

FLYING WITH ONE EYE

As one-eyed commercial pilot Frank B. Brady recounted in his must-read book A Singular View: The Art of Seeing With One Eye, in 1992 the F.A.A. reported that it had licensed 218 first class, 739 second class, and 2623 third class medical certificates to one-eyed pilots.

To fly as a Private Pilot, all you need is a 3rd Class Medical. Commercial (non-airline) pilots need a 2nd Class Medical. About the only people who need 1st Class Medicals are airline pilots. The FAA has created a new Sport Pilot category that may not require a medical at all.