On the Catholic Perspective on the incarnation of Jesus, according to St.Thomas Aquinas: Two natures, one Person To understand St.Thomas theology, we must understand the metaphysics that goes hand in hand with his theology.

For the notion of Person, the previous notion of Suppositum or Hypostasis seems to be central. I'm reading a book on St.Thomas' metaphysics, and it states that the difference between the concept of essence and suppositum was very important in how he was able to 'explain' the Incarnation of Christ, and His dual nature, of man and of God. 

How was St.Thomas able to explain the dual nature of Christ?
Check out at least these The Thomist articles:(note: the following is not an exhaustive listing of all The Thomist's articles on this topic)


"Aquinas on Nature, Hypostasis, &amp; the Metaphysics of the
Incarnation"
"Albert the Great &amp; Thomas Aquinas on Person, Hypostasis, &amp;
Hypostatic
Union,"
and especially:Jason L. A. West's "Aquinas on the metaphysics of esse in
Christ"


West also translated part of St. Thomas's Disputed Question: Concerning the Union of the Word Incarnate (De Unione Verbi Incarnati), which is a must-read for St. Thomas's view on the hypostatic union.

St. Thomas addresses the question of Christ's one being and two natures in De Unione a. 4 on "Whether there is only one being in Christ?":


  Objection 1: For there is both a divine and a human being in Christ;
  which cannot be one, because being is not said univocally of God and
  creatures. Therefore, in Christ there is not only one being, but two.
  
  ...
  
  I answer that ... Just as Christ is one simply on account of the unity of the supposit, and two in a certain respect on account of the
  two natures, so he has one esse simply on account of the one eternal
  esse of the eternal supposit. But, there is also another esse of this supposit, not insofar as it is eternal, but insofar as it became
  a man in time. That esse, even if it is not an accidental
  esse--because man is not accidentally predicated of the Son of God,
  as was said above [art. 1]--nevertheless, is not the principal esse
  of its supposit, but a secondary [esse]. Now if there were two
  supposits in Christ, then each supposit would have its own principal
  esse. And thus there would be a twofold esse in Christ simply. [Emphasis added]


St. Thomas also writes in his Compendium of Theology c. 212 on "the Unity and Multiplicity in Christ":


  For it is clear that parts divided individually have their own esse,
  but insofar as they are considered in a whole they do not have their
  own esse, but they exist through the esse of the whole. So,
  therefore, if we were to consider Christ as an integral supposit of
  two natures, there would be only one esse of him, just as there is
  also one supposit.