how to clean the outside of an briarwood estate pipe bowl

Cascade Plus . :rofl:

I normally use rubbing alcohol most times, but I'm not a fan of heavy stains. But if I'm trying to keep the stain I would use wet toothbrush or q tip (great for rims), then wipe with a fluffy rag. I know folks say briar and water don't mix and prolonged water contact can make the grain swell, but a little bit doesn't hurt it in my opinion.
You might also take a look at some stiff bristle shoe brushes. After that I do like Sam and use white diamond (unless it's got heavy rustication). You'll be surprised at the shine you can get from the polishing compound, cheap wheel, and a cordless drill.
I *just* started trying Carnuba wax on my last two restores. Not sure how I feel about it yet.
One tip I never see mentioned. Once you finish the clean up make sure you give the inside of the bowl a wipe down with alcohol. Lots of junk from the cleaning (briar dust, compound, wax) can get inside and give you a funky tasting smoke. I also never like how they smoke the first time or two. I can still taste the alcohol usually.
I'm still learning but there are some awesome refinishers here on this site. Show us some pics!

I run them under hot water and clean them with Murphey's .Next I use boiling water to break the cake and remove tars . Boil water and use a small funnel . Salt treatment .

Murphy's Oil Soap, undiluted, can/will remove finish and stain so be careful with that.
It all depends on how far you want to go of how little you want to do. Some cotton swabs and rags with a little alcohol and saliva can remove a lot of tar and gunk, with elbow grease. Buffing wheels can do great things, too. There are alternatives to carnauba wax, such as Halycon II and Paragon and other things like gun stock wax and clothes. Then there's soaking the stummel in a container of 91% isopropyl alcohol for 1-24 hours to remove all the finish and most of the stain. The possibilities are almost endless, depending on what you want to accomplish. But all of them have two things in common: patience and trial and error (okay, maybe that's three lol).

kgill. Yes, it will strip it. That's why I mentioned the water and brush/qtips.
And your 100% right about about cleaning it before your wax it.
I've used Halcyon wax before. I'm on the fence about it. You really need to use a TINY bit and make sure you rub it in very well. It dulls shortly after also imo.