Was the Son eternally begotten? Was the Son begotten continuously or just once?

This question revolves around the begetting of the Son before the creation of the universe NOT his incarnation.

Was the Son begotten continuously forever? 

Or

Was the Son begotten once forever?
One of the best (if not the best) Augustinian/Thomistic theologians is Frank J. Sheed.  His great work “Theology and Sanity” is an excellent guide for contemplating questions like these.

Sheed writes:


  What, then, is time?  Philosophers use the word in two closely related
  senses; most of us find that one of these is sufficient for us.  Time,
  say the philosophers, is the duration of that which changes; time, say
  the philosophers again and we with them, is the measurement of the
  changes of the universe. What is common to both statements is the
  relation of time to change.  Where nothing changes, time has no
  possible meaning. Thus time and the universe started together.  God is
  infinite and therefore changeless.  He is “the Father of lights with
  whom there is no change or shadow of alteration” (James 1:17).  He
  possesses the utter fullness of existence, so that nothing can go from
  Him, for He already possesses all.  The universe He created is a
  changing universe.  And because change belongs to it and not to God,
  time belongs to it and not to God. To repeat, time and the universe
  started together: time is the ticking of the universe.


If we say, “Jesus was begotten before the universe was created,” then we are stating something that has no meaning at all.  Before is a word of time, and there could be no time before the universe because time began with the universe.  To say “before the universe” means when there wasn't any “when”; which is to say that it doesn't mean anything at all.  The same is said about the phrase, “how many times was (was is also a time word) Jesus begotten.”  

Frank Sheed goes on to discuss processions in eternity:


  There may still remain one error clinging to our knowledge of the
  processions of the Persons in the Blessed Trinity because of our own
  immersion in time.  As far as the statement of it goes, we are not
  likely to make the error of thinking that the Son is in some way less
  eternal than the Father, or the Holy Spirit in some way younger than
  the Father and the Son.  We know that there is no succession in
  eternity, no change in God.  God the Father did not first exist as a
  Person and then become a Father.  God, by the very act of being God,
  generates his Son; God the Father and God the Son, by the very act of
  being God, spirate the Holy Spirit.  As I say, there is not likely to
  be any error in our statement of this: the error will tend to cling to
  our idea in such a way that when we are looking directly at it, we do
  not see it, yet it is profoundly there: and, because time is so deeply
  woven into all our experience, our advance in the knowledge of God
  depends upon our deliberate effort to rid our mind of it.


According to the historical Church's creeds and councils, the only valid answer is, "Christ is eternally begotten only of the Father."

The Nicene Creed was formulated by the Early Church Fathers to authoritatively define the  Christological absolutes that orthodox Christians hold:


  I believe in one God, the Father almighty,
      maker of heaven and earth,
      of all things visible and invisible. I believe in one Lord Jesus Christ,
          the Only Begotten Son of God,
          born of the Father before all ages.
      God from God, Light from Light,
          true God from true God,
      begotten, not made, consubstantial
         with the Father;
          Through him all things were made.
      For us men and for our salvation
          he came down from heaven,
          and by the Holy Spirit was incarnate
          of the Virgin Mary,
          and became man.


According to this formula, Christ is "the only begotten Son of God."  While being eternally "begotten from the Father before all ages,"  he was "by the Power of the Holy Spirit incarnate of the Virgin Mary, and became man." It is precisely because Jesus of Nazareth was eternally begotten/born (fully God and fully Man) before all ages, that the Blessed Virgin Mary can properly receive the coronation of being named "Theotokos," or Mother of God.