Being as it appears to be thermal, I would think electrical rather than carburetion (fueling) other than possibly running out of gas...! Have you tried turning the fuel petcock to reserve or prime if equipped, trying to start and idle immediately after it dies just to be sure? Sounds like you will have to limp home and start troubleshooting looking for electro-mechanical problems first, then with a heatgun (do not "melt" anything) last.
For EFI system you may want to look at electrical connections and wiring for cut or damaged insulation, signs of faulty connections, discolored or damaged contacts. Check fuses block fuse contacts for corrosion, and make sure that fuel filter in tank and hoses hidden from view are not leakng, the fuel pump and relay "stir", after engine dies when you try to restart.
If fueling is assured look at the Crank Position Sensor (CPK), Ignition Coil & Wiring, Throttle Position Sensor (TPS), Iginition Control Module (ICM) and finally Electronic Control Module (ECM) in that order (this is based on how I was dealing with my carbed Sporty intermittant problem). JMO