Brake cleaner like I said works very well at eating up carbon and built on gunk. I've used it for several years for everything from my wheel guns to M2 .50BMGs. However you need to make sure that you properly lubricate afterward as it will dry out your metal. All you need to do is spray liberally let it soak in for about a minute, scrub as necessary for amount of build up then repeat until you're all clean. After you have all the residue removed apply a coat of oil and rub it in with a rag.

I'm not an old timer but I've got a lot of time in the military, specifically in the infantry, and I've learned a lot of tricks for cleaning up as easily and as quickly as possible. Many of them I wouldn't recommend using on your Great-Granddaddy's SAA, but the brake cleaner won't hurt your stainless steel as long as you clean things up properly when you're done.

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Son of a Gun...Smith