A common thought experiment about shadows ask, what would happen if a bat flew across a very, very distant star? The farther the star, the faster its shadow would move, and there's no reason it couldn't move (propagate) much faster than light speed.
My question is, have we observed any real phenomena where something "moves" or propagates faster than the speed of light in vacuum?
How you define "phenomenon" is important. In the bat case, nothing is actually moving FTL - the bat is moving slower than light, the star isn't moving relative to us (doesn't end up mattering), and the photons carrying the information from the bat are moving at lightspeed. "Shadow" is a concept we use to describe the lack of light in a region, not a physical object.
Note that, for any observer whose light-measuring instruments are restricted to their immediate vicinity, the shadow just seems to appear and then disappear - it's just a flicker of the light. This is how it looks with local information. If you have many such observers and set them up in a big arc, and aggregate their recorded shadow-observations, you will see an apparent wave of shadow moving across the arc "FTL". But that information is nonlocal - you have to collect all the data and move it together after the fact, you can't know what every observer is seeing at any instant in coordinate time; being able to see that data as it's happening would require FTL communication, which is impossible because it involves sending information into the past (which of course can lead to catastrophic boostrap paradoxes). There wasn't actually any FTL motion, just a particular timing of local signals that when aggregated nonlocally yield an apparently-FTL result.
Consider also: to see the FTL shadow, you need to collect all the data afterwards in what would be considered a nonlocal way. If you try sitting at the end of the arc and having all the observers signal you with light beams (which is local information transfer) when they see the shadow, the shadow will seem to you based on those signals to be moving backward across the arc slower than light.
We have never observed any physical object moving FTL in any circumstance. Doing so would result in traversing spacelike curves which allow backwards time travel and all sorts of paradoxes, excepting the case of very particular spacetimes where the coordinate velocity of a null ray in a particular direction exceeds $c$.
