By what process did the miraculous conception of Jesus occur?

As far as I know, the previous answer is correct to say that the Church has no official position on this question. But here's a little evidence in favor of Christ being conceived from an egg cell of Mary. A prayer attributed to St. Thomas Aquinas, available at http://lordcalls.com/dailyprayer/prayer-before-the-reception-of-holy-communion , refers to "... the Body of Thine only-begotten Son, our Lord Jesus Christ, which He took of the Virgin Mary ...." (I don't know whether St. Thomas had more to say about this in his theological writings.)

John of Damascus thought the conception via the auditory sense.

The conception, indeed, was through the sense of hearing, but the birth through the usual path by which children come, although some tell tales of His birth through the side of the Mother of God. http://www.ccel.org/ccel/schaff/npnf209.iii.iv.iv.xiv.html

It's interesting because of the biblical "faith comes by hearing and hearing by the word of God".

In essence, the Church’s answer is, “We do not know,” and there is no way of finding out. There is no official Church teaching regarding this question.

(For a summary of the Church’s teaching on the Incarnation and what it entails, see Catechism of the Catholic Church [CCC] 456-483.)

There is some room for speculation here, based on what we now know about human reproduction and Christological doctrine. (Disclaimer: this is my own opinion, not the Church’s teaching, although I think it is well founded theologically and biologically.)

For example, we know that Jesus was a male (see, e.g., Lk. 2:7); this rules out the possibility of parthenogenesis (that the mother’s oocyte should be “fertilized” by another oocyte), since, evidently, that would produce a female child.

(For readers not familiar with human genetics, each of us possesses in each cell one pair of chromosomes—out of 23 pairs—that determines our sex; men always have a heterogeneous pair—an “X” and a “Y” chromosome—whereas women always have a h*m*geneous pair—two “X” chromosomes. In procreation, the mother and the father each provides one complement of 23 chromosomes—in particular, one X or one Y chromosome—which is why, by and large, offspring are evenly divided between males and females. Parthenogenesis is unlikely to produce a viable embryo anyhow, since there is a process called genomic imprinting that prevents chromosomes that come from the “wrong” gamete from functioning properly. Thus, an embryo possessing two maternal chromosomes in a pair would probably die before maturing.)

Although God could have created Jesus’ embryo without using any of Mary’s oocytes, it seems to me much more likely that He did do so, because God intended to partake fully of our human nature (CCC 467); and coming into being through the procreative capacity of one’s mother (and sharing her DNA, and so forth) is part of the human condition. (For example, it seems likely that Jesus resembled Mary physically, as is nearly always the case for children and their parents, and that would most likely have happened because Jesus shared her DNA.)

If that is the case, it seems reasonable to conclude that when the Holy Spirit “overshadowed” Mary (see Lk 1:35), that He miraculously fertilized one of her oocytes, providing the male complement of chromosomes.

The official Catholic doctrine on the matter is in accord with a commonly held belief among nearly all Christians that Jesus was conceived within a virgin, and that the process by which this occurred was a mystery and a miracle. The heresies of the Psilanthropists and Adoptionists are notable early exceptions, and modern criticisms of the doctrine come nearly exclusively from non-Christian sources.

ReligiousTolerance.org says it nicely,

The Anglican Communion, Eastern Orthodoxy, Protestant faith groups and Roman Catholicism have taught the "virgin birth," although the term "virgin conception" would be much more accurate. This has long been one of the Christianity's foundational beliefs, along with the inerrancy of the Bible; the God's inspiration of the authors of the Bible; the atonement, resurrection, and the anticipated second coming of Jesus, etc. All of the commonly used major ancient church creeds have also mentioned the virgin birth.

Specifically, the official doctrine of the Catholic Church is summarized in paragraph 497 of the Catechism of the Catholic Church,

The Gospel accounts understand the virginal conception of Jesus as a divine work that surpasses all human understanding and possibility: "That which is conceived in her is of the Holy Spirit", said the angel to Joseph about Mary his fiancee. The Church sees here the fulfillment of the divine promise given through the prophet Isaiah: "Behold, a virgin shall conceive and bear a son."

The Catechism also gives the following in the third statement of the Apostle's Creed, in reference to Christ,

He was conceived by the power of the Holy Spirit and born of the Virgin Mary.

Since the understanding of biological organisms and human reproduction did not include cellular phenomenon until the microscope was invented (the spermatozoid was first discovered in 1677 by Anton van Leeuwenhoek and the ovum in 1827 by embryologist Kart Ernst Von Baer) it is not possible for such a detailed scientific description of the event to have been posited until long after the canonical doctrines were established.

Even so, possible mechanisms of the virgin birth have been discussed in a paper published in Science & Christian Belief by R. J. Berry. The abstract from that paper states,

The Bible describes Jesus as being born to Mary ‘by the power of the Holy Spirit’, implying (although not stating) that the Spirit was his father. This has been rejected by some as an unnecessary doctrine, separating Jesus from the rest of humankind and dependent on an intrinsically incredible miracle. Such an objection is wrong: some form of distinctiveness like a Virgin Birth is theologically required if Jesus is to be divine as well as human, and there are several mechanisms by which the virgin birth of a male child could occur. The reason for recognising these is not to suggest that God necessarily used any of them, but simply to point out that apparent scientific difficulty should not determine the acceptability of a theological concept.

The paper not only discusses hypothetical mechanisms of spontaneous development of male children in human females, but also gives a nice overview of the development of the doctrine in the church and a history of its opposition, mostly from outside of the Christian Church. I expect that reading that article will complement this answer to address your question more completely.