Sounds like a fuel feed issue. IMO, there is a bad fuel check valve somewhere (fuel pump?) that's allowing the fuel to drain back when you shut down. Then when you crank it, it takes a few seconds for the fuel pump to replenish the fuel to where it's needed.
Larry, Debbie & Max the Pug
USAF Retired
2014 Itasca Reyo P
2016 Subaru Forester 2.5i 6 speed manual transmission