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:
- 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.
- 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.