When in doubt (or just as a rule), use Egg Replacer in cookies. It’s great and it works.
I imagine the flax seed “egg” would work well too, but the thing is, I don’t like nuts in my cookies so I don’t want the nutty, healthy flavor that flax produces.
Flax seed “egg”, equivalent of one chicken egg:
1 Tbls ground flax
3 Tbls hot water
Whisk and let sit for a couple minutes until a gel forms.