How do we "freeze" the display on an external display?

Use the extend mode instead of the clone mode.

Then, put whatever you want to display on the external monitor, and continue using the laptop monitor

click and hold fn, then F7. This will bring up the 4 options, click on the extend option. This will feed you projector a loop of what was on your screen when you pressed these buttons. To take it off of this, just press F7 again and select a different option, The small pictures pretty much explain everything.

Use extend mode, and drag the window which you wish to freeze to the side of your laptop screen. This should display on the extended screen.

In Windows one of the solution is using external display (as stated by each answer) but with some app for cloning/freezing, like ComroeStudios DisplayClone (it's available for free now) or UltraMon. Is is also possible to write your own app using GDI32 functions (PrintWindow, CreateCompatibleBitmap, ... and so on)