You can drill through a mirror using a diamond bit, but the process is tricky and should be done slowly. Here is a LINK describing it. This seems to me better done by a company specializing in glass and mirrors. Also you would need a rubber gasket between the pull and the mirror to ease pressure on the mirror.

There are some handles that attach to the back of the door and would not need to go through the mirror. These are called Finger Pulls. These pulls could be mortised into the edge of the door so the door fits flush when closed. These are made for kitchen cabinets so make sure what you buy is deep enough to fit the thickness of your door. Good luck.