I'm wondering this as well. My crazy neighbor cut mine down today to be a nasty mean spirited person. I found one larger stem in the little debris he left behind. I would love it if it could be rooted!

I've had good luck with green-wood cuttings. In the early spring, I cut them about 8" long, remove all but the top set of leaves and pot up in a small, 3"-4" container. Use potting soil and water daily. I leave the pots in the shade so the new growth doesn't get sunburned.

There are some good instructions at the link below.

Here is a link that might be useful: Propagate Rose of Sharon

I have very good experience with Rose of Sharon cuttings.
The best time to take cutting is when they are in dormancy .
Put some stem cutting on the edge of a small pot , Wet the potting mix and sealed inside a ziplock bag.
Put the bag in the shade and remember to ventilate every second day for some minutes. Don't add moisture until you notice new growth.
When they are not dormancy I am doing the same but the success is only 80%.
Ziplock bag is very useful when taking cuttings.

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I received cuttings in the mail during the hot weather from a member of this site. When I opened the package they were hot and sweaty. Almost all of them have sprouted and doing well. So they should be easy most anytime if a person needs to prune I would suggest they go ahead and try. I just used root-tone and put them in the shade when they started growing they found homes in my flower beds. When I grew them down on the coast it seems the soil was better there as they sprouted up almost any there the seeds fell, here in the Houston area they seem to not grow as well. I have the best results from cuttings over seeds but has sent them as far north as Michigan as I remember them doing very well when I lived there.