Super Fast Message

Good question! Imagine sweeping the bat signal really quickly across Mars. You could totally have it move faster than the speed of light across the surface!

But now imagine that you are standing on the surface of Mars while the bat signal is being swept over your position. You would like to send a message to another point on Mars faster than the speed of light. How do you send the message? To do so, you would need to modify the bat signal somehow (according to some prearranged code, such as Morse) as it's going over you to send your desired message. But you can't! Since you didn't send the light, you have no way to modulate the shadow so that it appears different when it reaches the destination. No one in the path of the shadow can change the shadow to use it to send a faster-than-light signal.

All you can do on the surface of Mars is watch the bat signal wash over your position and marvel that they have such a bright lamp down in Gotham.

On the other hand, if someone in Gotham wanted to send Batman a signal up on Mars, they could do so, but the light would take the usual $300000000$ m/s to get there.

shadows travel at the speed of light, just like light. if you were to create a shadow with the bat signal, that shadow would travel up to the clouds and back down to batman at the speed of light.

if you mean to imply that you could move a shadow across the surface of the moon at superluminal speeds, then you must consider that every instant contains a different shadow. in actuality, a series of similar shadows are being cast in rapid succession, much like a film is cast in a cinema. to convince yourself, consider the fact that you could cast the "same" shadow on both sides of the moon simultaneously using 2 projectors (or 2 separate templates on the same projector). for further reading on how to avoid this kind of identity fallacy, see the ship of theseus.