[embee]

If you do get soft carbon build up on the plugs it will "dry" but it's like a sponge so as soon as it next fires up from cold it soaks up the condensation and is way more prone to foul. To self clean effectively the plugs need to be up to something like 300C and running in a non-rich mixture regime, best achieved by allowing the engine to warm for at least a couple of minutes then holding the speed steady at around 3000rpm typically for another 30sec or so. That should be enough to be sure of avoiding repeated cold foul issues. It's not the rich fuelling per se, although that is what produces the carbon/soot, but the condensation of the water from combustion mixed with the soot which is the problem.

With any modern engine it is usually best practice to do starts "un-assisted", i.e. don't touch the throttle, and wait for something like 10sec after starting before using the throttle at all. Sure it isn't always essential, but that's usually the best way to avoid issues.
No need to run it until the fan starts, just until the plugs are hot enough, and rpm is the primary factor here.