If these symptoms only seem to occur after the petcocks have been off for a while, it sounds as if there is either very limited flow thought the petcocks and/or the Mikuni float chamber venting is restricted or the float needle is sticking.
Try disconnecting the petcocks from delivery pipes. Then open the taps and catch the petrol in a cup or something, observing the flow rate. Petrol flow is fairly gentle but judge for yourself if it seems to be sufficient. The seals inside the petcocks tend to either swell or break up after a long time and this either causes leaks or severe flow restriction. See if the symptoms are eased by having a full tank of fuel. If so, replace the petcocks.
The routing of the fuel tank pipes has a slight effect on the ability of the fuel to find its way into the carb float. Is it improved, for example. if both petcocks are opened? An air lock usually sits in the reserve fuel pipe. This should not matter unless the needle valve, for example, is a bit tacky and needs that little extra head of fuel to open it. Opening both taps would provide marginally more fuel pressure.
Check that the Mikuni float vent is clear. Failing all these, take the float chamber off and clean possible gums and distilates from the needle float area.
Have you had the Mikuni a long time or has this fault only just developed? What pilot jet size are you using? Presumably compressions and valve clearances are ok. Does it ever backfire of kick back? What ignition system is fitted?
My VM36 Mikuni always needs choke to fire from cold but once fired I have to flick the choke off immediately, otherwise it would be over-rich. It seems to be unable to fire cold without choke. The choke circuit is crude and non-progressive but does give the desired mixture to excite cold cylinders. I usually crank through 3 or 4 times slowly with choke on just to introduce some vapours into the bores. Then swing it with gusto and it fires first time. Then I flick the choke off and keep throttle tickled until it stabilises.