I found a lot of forum posts where people were asking how to replace the glass, but there were no good answers for any Hyundai model. I even found a few online service manuals, but they didn't show how to get the glass backing bracket out. As it turned out, it was very easy to replace, but I could only get a good look at the bracket once it was off the car. Since there was no information on either replacement option, I wrote up instructions for both.
Step 1: Order Parts
One place where I did find OEM glass for a good price was www.newhyundaiparts.com. Their price is $70.03 for the bare glass or with the mounting bracket.
I also found a whole side mirror replacement on Ebay for $58.40. Since I couldn't find good instructions to get the bracket out and it was by far the cheapest option, I went ahead and bought this one. This part is clearly not OEM, but there is no obvious difference between them but the mold numbers.
Step 2: Remove the Mirror Housing
If you have a trim removal tool, it is the best tool to use to remove the triangular interior trim panel. A screwdriver won't spread the force out enough and may mark the trim. You could use a putty knife if you have one with a rigid blade. In this case, there is a tweeter on the trim, so you should be careful about the wires in the back.
All you need to do is disconnect the wiring harnesses for the tweeter and the power mirror. Now you can remove the 3 nuts that hold the mirror on, I recommend removing the bottom 2 first, and then the top one. This will usually prevent the mirror from wobbling and scratching your paint. You may also want to have someone hold the mirror housing while you remove the nuts.
Step 3: Remove the Mirror Bracket
All you have to do is tilt the mirror plate all the way up until you see the white carriage behind the black glass backing. Insert the end of a flat head screwdriver into the long slot and twist the handle. This will pop the lower catch off without damaging any of the brackets. Once the lower catch is released, the bracket will be free to swing up and off the upper clips. Take your new mirror and connect the heater leads (if your car has heated mirrors), hang it over the top clips and snap the bottom in place.