Actually being browned is a mis-conception based on the originals which have aged - original rust bluing reverts to the red brwon stage over time.
I talked a while back with Don Stith about originals - Don has disassembled and closely examined more originals than anyone I know and is one of the most learned students of Hawkens there is.
Anyway here's what he had to say,
"The few Hawkens in good enough condition to see original barrel finish are blued. The blue is fairly dark. I like the color I get by doing a satin slow rust brown, followed by boiling the barrel in water to duplicate the color. When doing the rust blue, don't overpolish or you get too bright a color. It is not the deep translucent blue seen on European guns. The rib and thimbles were also blued.
The breech and tang were case hardened as well as the lock, butt plate, trigger guard, trigger bar,entry thimble and nose cap. Not the highly colored case of modern firearms but a mottled grey case that some of the early makers called forge casing. "
This is me...
For the mottled grey case color use Kasenit or to imitate the "look' without case hardening blue it then soak in bleach for no more than a minute or two at most. Wash thoroughly with hot water and soap, let dry, and then scrub back with dish scrub pad or if need be 4/0 steel wool. Repeat if need be to get the look. Once satisfied, neutralize with hot water and baking soda. Dry thoroughly and then oil. For this blue I use Borchwood Casey Super blue.
For the rust blue on the barrel, etc. Brownell's sells a couple of kits that include instructions. Or you can use LMF and boil in distilled water after each coat. For the last boil netralize with baking soda and water and then oil - old timers would use linseed oil while the metal was hot leave on for a few minutes and then wipe off the excess. You can finish up with a coat of wax or two.
FWIW - while the classic iron mounted Hawken Mtn Rifles are more "common", there are brass mounted original mtn rifles (not just their local small bore cheap rifles) still extant.