According to Catholicism and/ or Protestantism, how many times was Jesus of Nazareth, whether pre-incarnate or incarnate, begotten?

The Logos has two nativities begotten eternally from the Father before all ages and born of the Virgin for our salvation.

If anyone will not confess that the Word of God has two nativities, that which is before all ages from the Father, outside time and without a body, and secondly that nativity of these latter days when the Word of God came down from the heavens and was made flesh of holy and glorious Mary, mother of God and ever-virgin, and was born from her: let him be anathema.

Second Constantinople (553), Anathemas Against Three Chapters, Canon 2.

The first nativity refer to His begotten eternally from the Father while His second nativity is always referred to as born of the Virgin not begotten. Subtle difference but theologically significant. Because begotten refer to His eternal generation while born refer to His temporal conception by Theotokos. The same self Logos has two nativities He was begotten once in His divinity and born once in His humanity.

[T]his whole contemplation begins in the Scripture, because it begins with Jesus. Jesus is the Son of God. Well, how is he the Son of God? He’s the Son of God because he’s begotten of the Father, meaning he has no human begetter. He has no human father. His Father, literally, is God. And God who is his Father begets him before all ages, and then this very one, who is God’s Son, is born as a man, as a human being, from the Virgin Mary. And it’s interesting that, in Greek, the same verb, when it applies to a father, is “beget”; when it applies to a mother, it’s “born.” So we wouldn’t say that Jesus was “begotten” of Mary humanly; he was “born” of Mary humanly. But we would also not say that he was “born” of the Father; he was “begotten” of the Father. Those are important things to think about, and that’s why we’re talking about them now. We have to contemplate these things and get it straight, get it clear.

Fr. Thomas Hopko, The Holy Trinity.

The same Greek verb can be used in two distinguishable ways: one to designate the eternal relationship between the Father and His Son (μονογενής) and another to designate the temporal relationship between the Theotokos and her Son (γεννήθηκε).

Begotten of His Father before the worlds according to His Godhead; but in these last days for us men and for our salvation born of the Virgin Mary, the Mother of God according to His manhood.

Pope St. Leo the Great, Tome (448).

Once - in two different ways depending on our definition of 'begotten'

There seems to be some confusion on the subject because "only begotten" a theological term does not mean "begotten" a biblical term.

But to answer your question, if thinking 'begotten' as in 'only begotten' it gains prominence in Council of Nicaea in A.D. 325 representing Christ’s single eternal generation:

The phrase “only-begotten” is not an accurate translation and should not be used in any of the nine passages. This phrase is derived from the Latin Vulgate (a translation of the Bible from about the 5th century which has been quite influential on other translations) and reflects certain theological debates about the person of Christ. While the language of the Word being born of God is present in the 2nd century, the most notable occurrence of this language is the creed from the Council of Nicaea in A.D. 325. This creed speaks of the Son of God as begotten of the Father, unique—that is, from the substance of the Father—God from God, light from light, true God from true God, begotten not made, of one substance with the Father. This creed was the result of the rejection of the heresy that the Son of God was the first created being. Ultimately the phrase “begotten not made” leads to what theologians call the doctrine of the eternal generation of the Son. (Elwell, W. A., & Beitzel, B. J. (1988). Baker encyclopedia of the Bible (1590). Grand Rapids, MI: Baker Book House.)

Not to say that the theological term is not very biblical. When the word begotten is used with “son” it does mean God’s only Son, indicating God’s absolute delight towards him and his absolute uniqueness (Jn 3:16, 18; 1 Jn 4:9). The statement at the baptism and transfiguration of Jesus in the Synoptics 'This is my beloved son' also carries the same sense of God’s absolute delight towards him and his absolute uniqueness, but not principally a literal/spiritual 'birth'. Not 'principally' a 'birth' but by implication as Jesus is unique since he is also eternally begotten. If that does not make him unique I do not know what can. In fact the very notion of a son, and all its uses in scripture could be argued as having its ultimate meaning in dimly reflecting the glory of the Son. The concept of sonship finds its ultimate meaning in Christ.

To confuse matters more, in Hebrews Chapter 1:5 it reads ‘For to which of the angels did God ever say, “You are my Son, today I have begotten you”? Or again, “I will be to him a father, and he shall be to me a son”? this seems to imply a 'birth' or sorts, but not really. From this type of reference we are naturally tempted to mix the theological term (i.e. eternal generation) with the biblical prophecy first fulfilled in David (as a type) and in Christ as Messiah. Therefore, although “only begotten” is a theological term referring to a single eternal event, ‘today I have begotten you’ means something totally different in terms of a true generation, or 'birth'. This really refers to his 'birth' from the dead, or the first born.

John Owen explains the different views about 'begotten' very well here, as well as the correct view to take:

And some assign this to the day of his incarnation, when he declared him to be his Son, and that he should be so called, as Luke 1:35; some to the day of his baptism, when he was again solemnly from heaven proclaimed so to be, Matt. 3:17; some to the day of his resurrection, when he was declared to be the Son of God with power, Rom. 1:4, and Acts 13:33; some to the day of his ascension, whereunto these words are applied. And all these interpretations are consistent, and reconcilable with each other, inasmuch as they are all means serving unto the same end, that of his resurrection from the dead being the most signal amongst them, and fixed on in particular by our apostle in his application of this testimony unto him, Acts 13:33. And in this sense alone the words have any appearance of respect unto David, as a type of Christ, seeing he was said, as it were, to be begotten of God when he raised him up, and established him in his rule and kingdom. (Owen, J. (1854). Vol. 20: An Exposition of the Epistle to the Hebrews, Volume 3 (W. H. Goold, Ed.). Works of John Owen (136–137). Edinburgh: Johnstone and Hunter.)

Conclusion: The biblical term “begotten” refers to the resurrection of Christ, which happened once. The theological term “only begotten” refers to the eternal generation of the Son, by an inconceivable communication of the essence and substance of the Godhead by the Father --- only happened in a continuous forever. God’s absolute delight towards Christ and his absolute uniqueness implied by the words 'only Son' brings the single eternal generation and the single generation from the dead together under the Sonship theme. Both his eternal generation and his resurrection make him absolutely unique and proven fully approved of by God. This is partly why Jesus always used the phrase 'the Son' as implying being one and only person generated from yet equal to his Father, for example, as in John 5:16-29.