Hi,
This is how I do it with eggs.
First scrape the flesh side of the pelt with a dull drawknife type implement to remove any membrane and flesh. Underneath the pink membrane you will see that the skin is slightly mottled and this is the surface you want to get back to. Sometimes you can peel this membrane off but you do stand a chance of ripping the delicate pelt. Be careful at the edges where the skin is thinner or you'll tear holes in it. Also, drape your pelt over a round, smooth log to scrape it, securing it by pinching the top of the pelt between your mid section and the top of the log then pushing away from you with the de-fleshing tool.
Next, whip up a whole egg (or the brain of the rabbit) in a bowl with a little warm water and tip it onto the flesh side of the pelt, rubbing it in for as long as possible. If you've got time, fold the pelt over flesh side to flesh side and leave it somewhere cool for a while, maybe even overnight. Don't leave the pelt damp for any longer than that or the hair may start to slip.
When you come to dry the pelt, pat it with an old tea towel to soak up any excess moisture. It should feel cold and clammy but not slippery. Slippery means that it's retaining too much moisture and will take ages to dry fully so keep soaking up the moisture with the towel. Now you have to work the pelt as it dries. Sit somewhere fairly warm - next to the campfire or woodburner, but don't force dry the pelt or it'll come out hard. Using fingertips pull and stretch it side to side, top to bottom, they're pretty stretchy at this point. After a bit you will notice some areas, when stretched will whiten. Feel these bits - they should feel cool and soft but not clammy like the rest of the skin. As you work, more and more of the skin will turn white - you need to achieve this white, soft look all over the pelt. Any thicker areas can be rubbed back and forth over a sharp (ish) wooden edge like the back of a chair. It's quite a lot of pulling and stretching but the time soon goes when you're chatting or listening to the radio. As long as every fibre is stretched as it dries, your pelt will be lovely and soft just like the chemically cured ones.
Then smoke it to 'set' the work you've done and ensure that the pelt stays soft if it ever gets damp again. Do this by suspending it up in the smokey eaves of your camp shelter when next out in the woods or make a proper smoker of some sort. Hardwood chippings mixed with a little punky (rotten but dry) birch is great. The longer you smoke it, the darker the flesh side will go. A tan/buff colour is about perfect but any amount of smoking is good.
Good luck and a Happy New Year to you!