Copy and paste from another forum, this guy seems to have figured it out.... I found this while trying to track down this same exact start-die-start-run perfect issue and P1351 code on my 99 2dr Tahoe
"The situation you described fit a faulty check valve in the fuel pump.
This would allow pressure to bleed back into the tank, and ultimately ,on a warmed up engine , cause the fuel line to go dry. Leaving only the fuel at the spider.
Then when restart is tried, the pump pressures up the system, but has a huge air pocket, however makes pressure nonetheless . That pressures the fuel laying in the spider and the truck starts, but then immediately dies because the vapor in the line ,under pressure has reached the spider, and has to bleed out the injectors and the return system. Enough vapor passes through the injector to stall the engine, and bleed the vapor off , then eventually pushing fuel back into the spider and to the tnk Via the return system.
This would set a p1351 because the Electronic spark timing would be active and the engine would be lower than 250 rpm because of the stall."
Check fuel pressure bleed off