Is it possible for an investigator to finish all actions and then have the butler provide an additional action to that investigator?

Example

Investigator 1 finishes all actions

Investigator 2 finishes all actions

Butler uses his special action to give an action to investigator 1

Investigator 1 uses Butler's action

Yes. He can. That's really the biggest benefit of his special ability.

What makes you think he can't?