@reidbarber If you’re not going to go through the mirror and into the wall, you have a couple options. One is you can have brackets that attach to your monitor and hold the mirror in place, top & bottom. These would tie the glass to your monitor and hold it in place by clips on the edges. Unfortunately, it means you can see the clips on the edge of your frameless mirror.
The other alternative is to mount some sort of bracket on the back of the glass that can be hooked over the top of the monitor. Without the bezel, my monitor is about 3/8" thick at the edge. A bracket that is roughly the width of the top edge of your monitor (in whatever orientation) that is essentially a u-shaped channel might do it. You’d want to attach the channel to your glass with a strong epoxy. At that point, the glass would just hang over the top edge of your monitor.