I had precisely the same problem with my 86 FJ60 last summer and it always happened at the worst possible time, e.g., on a tight, steep switchback at 12900' on the Argentine Pass road with 800 vertical feet of nothing but clean Colorado air immediately behind. Exceptional independent Toyota mechanic couldn't figure it out, so here's what I did (not sure what, if any, of this fixed it but it hasn't happened this summer):
(Guessing fuel related):
Had carb rebuilt (float level was okay).
Repalced fuel filter (twice, actually...ran my 40 gal. aftermarket tank way too low on gas just before the stalling started occuring and subsequently found what looked like big flecks of silver paint in the filter...might have been just a coincidence but the stalling began on the first trail ride after running very low on gas).
Checked fuel pump (was working properly).
Replaced gas cap (gasket worn out).
(Guessing vacuum related):
Not having any idea otherwise of where to start given the maze of vacuum hoses, replaced all of those little white valves which connect vac hoses at various and (to me) inexpilcable places.
(Guessing electrical related):
Opened and cleaned up all of the fusible links (I think that's what they're called).
Tightened up the screws securing the ignition cylinder in the steering column (had been having to jiggle key to insert).
I'm no mehcanic (obviously), but it doesn't stall now (knock on wood).
Does yours start right away after it stalls? Mine usually didn't; had to get it pointed downhill and rolling while cranking to get it to fire.
Good luck.
By the way, I bought it new in Aug. 86 and rolled over 320k on the way home from the office Monday; and I'm heading to Mexico's Copper Canyon in it next week.