Check ur that lines r clear, make sue the cap is vented, see if it does it with the fuel cap off. How hot is the engine getting on the gauge, if it works?
After that simple stuff, check timing.
That was my first thought also wild, maybe a older tank with some trash in it. But if he's getting enough fuel to restart using more throttle there must be fuel getting to the bowl.
I think your on to something with the ignition.....if you've got high heat conditions underhood a weak or failing coil might be the culprit after it heat soaks?