Does anyone knows how to a sign which monitor to be the full screen.

Usually you would address that issue from within the graphics card software.
For example, with nVidea cards, the nVidea Control Panel contains all of those settings.
Then, within PPro, in the Playback Settings you need to be sure the external monitor is identified and turned on.

Sorry I don't have the software in front of me at the moment, but it is in the Edit>>Preferences menu. There's a simple dialog to choose your default output.