Best bet is to let EGT stabilize. I typically let the EGT on my truck stabilize below 400F so it does not cook the oil in the bearings. Turbo cores are oil, water or both cooled units.
Unless you pull a long hill at 1000f plus temps and come to an abrupt stop at the top shutting the truck down with elevated output pressure, the turbo starts cooling immediately after the pull. I have monitored mine and usually notice even after hard driving, it takes less than 1 minute after parking to stabilize below 400F.
If you are concerned, watch EOT along with EGT. EOT while driving will be 20-30F above ECT (flat driving pulling trailer), when you stop, EOT's usually drop below ECT's. Watch EGT, EOT, and ECT to see when they stabilize. Once the stabilize, wait 30 - 60 seconds before shutting the truck off.
I think the extended run feature has to do with the regen cycle. If it does not complete a regen within X number of shutdowns, it will restart and finish the cycle.
2004.5 Silver Birch Metallic GMC K2500HD
18 Denali CCSB 2500 4WD Mineral Metallic with Cocoa/Dark Sand Interior