What did I do?

I could see the moon clearly (with the face) through the viewfinder, but could also see the moon face "shadow". But in the shot on the camera, I'm only seeing a white circle for the moon, and clearly see the "shadow". Never seen anything like that before. My thought is I just over-exposed for the moon, which is why it's just white, but anybody have any thoughts on where the "shadow" came from?

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Post your EXIF data.

Aperture too wide, shutter too slow or ISO too high (or combination). You want to set your exposure mode to spot then place the focus point over the moon.

If in manual mode, start with:
f/11
1/125 - 1/250
ISO100

...and adjust accordingly.

Really like the flare

The moon is one bright beast!!!

You need to way underexpose to get the surface features of the moon.
Is the ghost a sort of lens flair? I'll leave more experienced guys and gals to answer that for sure...

The trees are probably screwing up your camera metering. I had the same thing happen tonight. Take a reading just on the moon itself and then widen the POV. I would kick it down a stop from there as tmcman said you have to underexpose.