Why is Jesus Christ only one person despite having two minds?

Questions like this one concerning the incarnation of God really stretch the limits of what we can comprehend. Over the centuries Christians came to settle on terms like "person" or "hypostasis" as the best ways they could think of to explain and reconcile the many truths taught by the scriptures, even though they can't do justice to the full reality of God. But I don't think we should feel hopeless: rather than being completely transcendent and unknowable, God wants to be known by us, and so he has both equipped us with intellects that reflect his own, as well as revealed himself to us in the scriptures and in the incarnation of his Son, Jesus. We can think of many analogies which, although not perfect, can communicate true things about God.

So one analogy I find helpful for understanding how the one person Jesus Christ can have two minds and two wills, is the fact that all of us actually do too, in a way. The two hemispheres of our brains work closely together, but if their connection through the corpus callosum is severed, a condition called split-brain arises, and it is really weird. For a quick introduction I'd recommend this five minute Youtube video by CGP Grey. People with split-brain can experience the two halves of their brains making different decisions, having different thoughts, and even having different favourite colours. This capacity for each hemisphere to make decisions doesn't arise after the severing of the corpus callosum, it was always there, but in a healthy brain the two hemispheres cooperate so well that we don't realise our two hemispheres could operate independently. Similarly, in the incarnate Jesus, his two natures cooperate perfectly and each nature's faculties of thought and will are united. I think even the passages where Jesus describes his own ignorance, such as Matthew 24:36 where he says he does not know the day of his own return to earth, don't indicate a division between the natures, but instead just a holding back of certain facts from the divine nature to the human nature.

What actually is a "person" of the godhead? I think we should understand this as referring to the relational distinctions within the godhead. There are three "persons" of the Trinity because there are three loving relationships: the father and son, the father and spirit, and the son and spirit. Even though there are more than three aspects of God, and even though aspects like the will of God are taught by Chalcedonian Christians to be singular and shared by all persons of the godhead, neither the multiplicity of aspects nor their shared nature are responsible for how many persons of the godhead there are. The "persons" of the Trinity refer to whatever it is that loves the others. We recognise this in humans too. Even though you could say that you genuinely love both the body of your spouse and the mind of your spouse, we recognise that there's a love for a person that unifies and subsumes both. So it is with the incarnation: even though God the Father could say that he loves both the human nature of Christ and the divine nature of Christ, he loves the united God-man Jesus Christ as one person, just like any of us are one person, not two.

There is a complex Christological history behind this question. The common ground is that Jesus Christ is of two natures, divine and human. Christ's divine nature is uncreated and pre-existent, while his human nature is given through the Blessed Virgin Mary. This was then taken in two ways:

1. The theologians of Antioch in Syria emphasized the completeness of the two natures of Christ and his being fully divine and fully human, and that meant that in Christ incarnate there are two wills (or minds). This is dyophysitism. In order to be fully human, Christ needs to have a fallible human will, by which he can be tempted, otherwise tempted, fallible human beings cannot be saved through him. This is alongside Christ's perfect divine will. Nestorius, who became Patriarch of Constantinople in 428, was trained in Antiochene theology and was appalled by the popular devotion to the Virgin Mary in Constantinople by which she was addressed as Theotokos (or 'God-bearer'), believing that she should only be credited with the producing of Christ's human nature. This made Nestorius unpopular and led to accusations that he was advocating that Christ was two persons.
2. The other view came from the theologians of Alexandria in Egypt and was particularly expounded by Cyril of Alexandria, who became Patriarch of the city in 412. The Alexandrian view is that a person can only exist with a unified nature, and so Christ is of two natures, but, at his incarnation, these became one nature. This is monophysitism (but modern-day supporters of a moderate version of this view prefer to be called miaphysites). Thus, Christ has one will, even though that will has a fallible human part as well as a perfect divine. Cyril affirmed that the two natures were in united in Christ (a 'hypostatic union'), but others went for more extreme views. Apollinaris of Laodicea (died 390) advocated a view in which Christ's human nature is 'hollowed out', leaving a shell that is filled with his divine nature at the incarnation. And Eutyches of Constantinople (died around 456) taught the one-nature Christology so strongly that Christ's human nature was completely subsumed by his divine nature. The extreme versions of monophysitism tend towards an understanding of Christ as a kind of superman, a tertium quid who is not really wholly human anymore.

For a time, Cyril of Alexandria had the upper hand and Nestorius was exiled, but extremist one-nature Christologies needed to be rectified. The pendulum swung back, and the Council of Chalcedon (451) promulgated an essentially moderate two-nature Christology with Cyril's hypostatic union. This is the orthodox Christology of Catholics, most Protestants and the Eastern Orthodox churches.

There was, however, a strong monophysite dissent to Chalcedon, leading to centuries of attempts to find a different definition, which essentially came to nothing. One attempt at reconciliation was the seventh-century doctrine of monothelitism that taught that, whereas Christ is two natures, he has but one will. This doctrine secured official support in the East for a time, but, in the end, was condemned by the Third Council of Constantinople (680/1) for diminishing the fullness of Christ's human nature in order to reach a political compromise.

While the Christological debates do not say much about 'minds', the sense of 'will' comes closest. Christology is traditionally done working from the point of Jesus Christ being both fully human and fully divine and then navigate between the extremes of the two schools of thought. We might like to start from our experience of being human, which leads to the prevalence of new ways of expressing what are basically either adoptionist or Eutychian views, or even whatever Neo-Apollinarianism really is. An approach from human experience will also prefer an essentially monothelite response because, otherwise, we are looking at schizophrenia. The patristic approach, however, has been to ensure the fullness of the two natures, while disagreeing whether the hypostatic union makes them into one or holds them as two still. Perhaps the two approaches find common ground in some kind of Cyrillian miaphysitism that ensures the fullness of natures in perfect unity.