Or, I should say, he's not actually chewing on the screen itself, but rather he seems to be methodically going over the screen's surface, picking up whatever bits of food are on there. As far as I have been able to find out, katydids are primarily plant eaters, so I have no idea what this creature is finding to eat on the screen. Whatever it is, though, it must be really delicious. (I also once noticed a katydid doing the same thing to the windowsill, so maybe there's something interesting all over the outside of our windows, and the screen's just the easiest surface to cling to.)
In addition to being totally perplexing, this behavior has also has given us a fantastic opportunity to watch a katydid's mouth at work, from right up next to the katydid's face. And I've learned an important fact: Katydid mouths are weird! There are so many mouth parts! Also, these creatures are just really cool to watch.
So if you want to see a katydid's mouth in action, here's a video I took of this creature on Friday night. (Thanks to Paul for letting me use his camera.) We've named the katydid "Karl," although of course I have no proof that it's the same insect visiting our screens on multiple nights.
Interestingly enough, in doing a quick search for "katydid on screen," I found another YouTube video of the same behavior. So our katydids aren't the only ones doing this. I'd love to eventually find out why this is going on, but for now I'll just stick with the "hey, cool" reaction. Insects are so crazy!