Modern Vespa

Cold start, starts. Give it gas - it dies. Start it again, electric start, no issues giving gas, but first try dies with gas… confusion.

I had this problem with my Vespa right when it was brand new. I think in my case the problem was that I just overfilled the gas tank - most likely there needs to be some air there for the whole system to function properly. After restart it also worked fine. If you've recently filled your tank, try filling it a bit less next time.

And for a good measure try a different gas station.

I had what sounds like the same problem with my Piaggio Liberty last year... It was still under warranty and due for a service anyway. I figured my evap system was boggy, but it turns out my spark plug was in desperate need of replacement. Have not had any stalling issues since.