This sounds like a vapor lock/hot fuel issue. I would check for spark as was suggested to eliminate the ignition being the problem. You can get a spark plug tester for $12 or so at any auto parts store. Get the car good and hot to where it cuts off and then use the tester to check for spark.

If this is a fuel issue like vapor lock, you will need to get a fuel pressure gauge (0-15psi range) and plumb it in with a T-fitting on the fuel line or use the handy plug you have on your dual fuel lines. I suspect that when your car "shuts off" its fuel pressure if dropping to near zero and the motor is starving.

Get a spark plug tester and a fuel pressure gauge and do some testing. Been there, done that, got the tee shirt.