I like cutrightjim's suggestion of replacing the interior portion of the ceiling board. You could completely eliminate the primary problem this way.
Otherwise, 3/8" is nothing to a good taper (or an amateur with patience), as long as you have room to work it out. You do. You'll need a wide knife or trowel, though. 24" minimum.
Fill the void with drywall to the level of the lower plane (higher with respect to the floor). Skim between the planes to pre-fill, and allow this to dry completely. It might take several days. It should be completely white throughout.
Now tape both sides of the channel as normal. Once that's dry, start leveling. Apply compound from the edge of the higher plane outward, as wide as your trowel. Again, let that dry thoroughly. Again, it might take several days.
Continue that process, using the low edge and the contact point of your trowel as screed guides. Finish by skimming the whole area, tapering gradually over 32" or more. Only light sanding should be needed if you've done a good job.