Sometimes left, sometimes right … wow. If Rod Serling were still here, he’d say “welcome to the Twilight Zone”.

I agree, make sure your spark plug cables are good. I’m no electrical guru, but seems like some kind of overall energy issue. Electricity does strange things. If the primary voltage is borderline too low, maybe that would cause random issues in the secondary when cold. I’m only guessing. If you have an inductive timing light, perhaps that could be used to see if ignition is present or absent when trying to start.  Would be really cool if you had two lights hooked up with triggers pulled, then try to start.  Could see who is firing. I’d probably power the lights from a separate 12v source.  Be good to catch it in the act, if it is an ignition issue.  Would also be interesting to swap ignition modules.   

Great mystery on your hands!

So a bit of analysis using a timing gun reveals no spark on LH cylinder for several revolutions.  I only have one gun, so I'll try the RH cylinder in the AM.  My theory is the resistors in the Hall sensor are dropping resistance when it's cold, just enough to make it not fire until things start cranking a bit.  Could be completely wrong, but if true, it would explain the randomness of one vs the other cylinder firing, as well as the fact it starts right up when temps are warmer.  Going to pull front cover, check bean can plug (described by Richard above), and go from there.

Very good, you “caught it in the act”.  I’ve never had a bike newer than ‘78, so I know nothing of the BMW hall sensor/bean can system. Keep us posted!