Why Do Catholics Call Mary the "Mother of God?"

Lately we've had at least one or two protestants who believe the Blessed Virgin Mary was not the "Mother of God" and that it is blasphemy to call her that.  While they are right that God had no beginning, God the Word became flesh in time and space and became one of us.  I would agree with their objections if Jesus Christ were just a man.  He's not.   They, however, are falling into the same trap as Nestorius and his followers--that God was only residing in the body of a man; that He was two persons as well as having two natures.  St. Cyril sent at least three letters to Nestorius on this very subject, below I quote from the third, which was read at the Council of Ephesus, 431 AD.  [Quoted material in purple]

For we do not divide up the words of our savior in the gospels among two hypostases or persons.  For the one and only Christ is not dual, even though he be considered to be from two distinct realities, brought together into an unbreakable union.  In the same sort of way a human being, though he be composed of soul and body, is considered to be not dual, but rather one out of two.  Therefore, in thinking rightly, we refer both the human and divine expressions to the same person.  For when he speaks about himself in a divine manner as "he that sees me sees the Father," and "I and the Father are one," we think of his divine and unspeakable nature, according to which he is one with his own Father through identity of nature and is the "image and impress and brightness of his glory." But when, not dishonouring the measure of his humanity, he says to the Jews:  "But now you seek to kill me, a man who has spoken the truth to you," again no less than before, we recognise that he who, because of his equality and likeness to God the Father is God the Word, is also within the limits of his humanity.  For if it is necessary to believe that being God by nature he became flesh, that is man ensouled with a rational soul, whatever reason should anyone have for being ashamed at the expression uttered by him should they happen to be suitable to him as man?  For if he should reject words suitable to him as man, who was it that forced him to become a man like us?  Why should he who submitted himself to voluntary self-emptying for our sake, reject expressions that are suitable for such self-emptying?  All the expressions, therefore, that occur in the gospels are to be referred to one person, the one-enfleshed hypostasis of the Word.  For there is one Lord Jesus Christ, according to the Scriptures....

Mary, Mother of God, with her Son

Therefore, because the holy virgin bore in the flesh God who was united hypostatically with the flesh, for that reason we call her Mother of God, not as though the nature of the Word had the beginning of its existence from the flesh (for "the Word was in the beginning and the Word was God and the Word was with God," and he made the ages and is coeternal with the Father and craftsman of all things), but because, as we have said, he united to himself hypostatically the human and underwent a birth according to the flesh from her womb.  This was not as though he needed necessarily or for his own nature a birth in time and in the last times of this age, but in order that he might bless the beginning of our existence, in order that seeing that it was a woman that had given birth to him united to the flesh, the curse against the whole race would thereafter cease...

1.  If anyone does not confess Emmanuel is God in truth, and therefore that the holy virgin is the mother of God (for she bore in a fleshly way the Word of God become flesh, let him be anathema.

2.  If anyone does not confess that the Word from God the Father has been united by hypostasis with the flesh and is one Christ with his own flesh, and is therefore God and man together, let him be anathema.

[Third letter of Cyril to Nestorius, read at the council of Ephesus and included in the proceedings.  Two of the twelve anathemas proposed by St. Cyril and accepted by the Council of Ephesus are included here.]

This is the whole point of the Church calling the Blessed Virgin Mary the "Mother of God."  Our Lord Jesus was one person with a divine nature and a human nature.  He was not two separate persons residing in one body.  Just as man and his soul are one and inseparable, so God and man in the person of Jesus Christ Our Lord is inseparable.  He is God.  If you believe that Jesus was and is God, then why would there be an objection to calling Mary, His mother, the Mother of God?

[Added 8 Sep 2010, 9:45AM:  Here is an online source of a translation of the documents of the Council of Ephesus, 431 AD:  http://www.ewtn.com/library/COUNCILS/EPHESUS.HTM  It looks like the same translation as the hard copy I quoted from here.]