Just a thought, but this could be related to the security built in to the key. It will cut the fuel and thus cut off. I am doing this mobile so I can't tell what year your vehicle is to know if you have the security key. You may try a spare key and see if it does the same thing. Just a thought.

 

I've heard of this security feature... My key broke off in the ignition about eight months ago and I was able to find a new spare that used my same "chip".... it has been eight months now so I dont think this is the case... Ive also narrowed the stall down that it only accurs while the engine is still hot.... never happens in the mornings... but if I turn the jeep off, run into the store, start it up...... stall almost every time

 

If it was the key or the security system a small red light over the temperature gauge will flash a few times..at least that's what my 07 does..I found it the hard way after I picked it up used from the dealer...take it home, no problems, go to a buddies house to show off the new ride, and it would crank, fire then stall...it took a few times of me trying to start it before I noticed the little light flashing...after trying anything I could think of...checked fuses, checked battery terminals, had a friend tap on the fuel tank while I cranked it over...finally I called the dealer and he said"oh yeah that happened to us too...have you tried the other key?"sure enough the previous owner turned it in with only one chipped key...the other was a copy just to get in.
Since you broke your key in the cylinder a while back and your symptoms are exactly like what mine where..I would bet you need to have the key pieces either removed by a locksmith somehow or replace the lock cylinder...not sure how much of a pain it its on the jk or how the electronics for the security system are tied in

 

Just a thought, but could it be that a fuel line is close to the exhaust and when the fuel stops moving (when you shut it off) that it heats the fuel up, expands, and causes a small vapor lock which in turn causes a misfire. If you have a friend that has a mechanics computer and can leave it hooked up while he rides around with you and try to simulate your driving habits and see if he can read the misfires on the computer. A misfire can go undetected from the jeeps computer if it doesnt last long. It could run the gambit of injectors, coil pak, stopped up converter. Hope this helps and i hope you fiqure this out.

 

Have you changed the fuel filter recently? Likely a fuel pump or restricted fuel flow issue.