An external removable frame/bracing structure is one way to support the mirror. Making hidden mounting brackets that are attached to blocks (a variation of the toggles you have) that are attached to the mirror with mastic (the technical term for mirror glue) is another way.
There are several schemes depending on size, appearance, and shape. The mirror can be drilled to have supports that go through the face of the mirror.
The simple way is to get the glazier people who cut and polish the mirror to hang it! Then if they break it, they just go make another one and you do not have to pay for two!
I would recommend using a cement board type of drywall since behind a mirror can get very damp. This type of board is much better than green board. It is the same kind of thing you use for tile in a wet area. Totally waterproof and can even be obtained with reinforcement like wire or fiberglass strings in it.
If you want more details on DIY let me know.
Be safe and happy remodeling!