Q:

What scientific principles could be 'abused' to make the moon have physical phases?

The lunar phases describe the form we perceive the moon having based on the light it reflects from the sun while being partially turned away from us.
Q: Can we abuse science to make the moon have physical phases?
By having physical phases I mean that the physical shape of the moon changes in cyclic manner. Thus if one would land on the moon, in its crescent the moon would be an actual crescent opposed to just appearing as one due to what we see reflected.
We can do the following things:

change lunar composition
change its orbit
make it rotate
massive alien engineering projects

We can not do these things though:

change the mass of the moon

A:

Changing your moon's shape
Make it look a bit like a really fat crescent roll. And slightly more crescent like as well. This would mean that the center part should give a circle shaped view while if you saw it from the side, it would look like a crescent. Now just have it rotate as per normal and voila.
So it's not exactly having your moon change into a crescent and then back into a sphere. But it might be the most feasible one. And then you can hand wave the fact that someone took a real large bite out of your moon.

