What blocks the moon when it is half a moon?

The object that blocks the moon when there is a half moon is the moon itself. The moon is a sphere, so only one half of it can be sunlit at any moment. The other half is dark just because the sun is below the horizon for that half of the moon. And when we see a half moon, we are simply looking at the moon just as the line of sunrise/sunset cuts across the moon's diameter.