Which theory is supported most by the finding that the moon is similar in composition to the outer portions of the earth?

The "giant impact" theory.

The most popular current theory of the Moon's formation is that during the early times of our solar system very shortly after its formation, another planet perhaps as large as Mars collided with the proto-Earth. The cores of the two planets merged (resulting in the unusually massive iron core of the Earth) and the debris blasted into space by the collision partly fell back to Earth and partly formed the Moon.