Hmmm... I don't recall that story from the Bible.
Lots of items and canonical books written at the time didn't make it into the Bible...there are some in the apocrypha you've probably never heard about either.
Judging from Adam's reaction, he knew right away that Eve was a part of him. That, in fact, was why God made Eve from Adam, so they would be compatible and attracted to each other. One of the sillier Hebrew traditions explained that by claiming that originally God created the first woman from the ground, the way he did with Adam. Her name was Lilith, and she was even more beautiful than Eve. But she did not want to be dominated by man, so she ran away. Adam complained to God that the wife He had given him had deserted him, so God sent three angels to bring her back. They found Lilith in the Red Sea, and they warned her that if she did not return, a hundred of her demon children would die every day. But Lilith preferred this punishment to living with Adam. Therefore, God decided to avoid repeating His "mistake" by creating the second woman from the body of the first man, for "only when like is joined unto like the union is indissoluble." The legend ends by stating that many demons descended from Lilith still lurk in the world today (it doesn't say how she got the children, though).