While I've never extracted brains for food,
I have boiled and cleaned skulls.   I'd probably
use a bone saw or hack saw to cut open the
skull instead of a hatchet.  I'd also skin the doe
head first.  That might make it easier for you since
it will then look less like a soft&furry bambi.  Of course,
the eyes will be bulging out, so maybe that would
make it worse.  I'm not at all squeamish so I can't make
that call.

Anyway, if you want to see a very interesting procedure,
you should watch a hippo being butchered...

(Hippos are delicious, BTW :)

Julie