Hi,
The adult moths are safe to feed to your Beardie. As for feeding the moth, you will first need to contain in it, preferably in a large cage. They are very strong fliers and will beat up their wings @ night trying to fly off. My suggestion would be to feed to beardie as soon as it ecloses (comes out). I wrote an article, long ago, for a chameleon e-zine & in it I have pics of feeding a crippled moth, you can try similar. They have a very long proboscis (tongue), which can make it difficult to feed in captivity to the inexperienced - even for the experienced!
http://www.chameleonnews.com/05DecCaruthers.html
@ scroll down to
ECLOSION
When making "nectar" use 4 parts water to 1 part sugar.
Another option is to put it into container in fridge (not freezer!) to temporarily chill him & make it easier to handle. Just offer to beardie.
Good luck!