What would cause headlights to shutoff by themselves?

Start by looking for a defective head lamp relay in the relay box.

Check the headlight relay in the power distribution box under the hood.

I love a mystery!

Sorry, I was on the page for concealed head lamps.

The next time it acts up, try the high beams, and the flash-to-pass feature to see if the high beams work. Two different circuits feed the multi-function switch. One is for the flash-to-pass feature and the other is for the high and low beam circuit from the head light switch.

If the high beams also cut out intermittently, go back to the head light switch and check for any terminals that were overheated previously. Usually the plastic connector body will be melted too around two terminals. Those blackened terminals will continue to cause trouble even when the main cause was the old switch and it has been replaced.

If the terminals look okay, locate the 14 gauge light green wire in the switch connector and monitor the voltage there with the head lights turned on. Whether that voltage disappears or not when the head lights turn off will determine which way we go next.

Also notice how long it takes for the lights to turn back on by themselves. If it's always about the same amount of time, that would be typical of the built-in circuit breaker kicking out due to excessive current. If the times to reset are random in length, suspect a poor connection or corroded splice as the more likely cause.

Excessive current will be a greater factor with the engine running because system voltage will be higher causing more current to flow. That might provide another clue.