Does Judaism view the Messianic era as a supernatural time?

Well, there definitely are sources that would imply so. According to the Midrash, in the Messianic era, plants will yield their produce on the same day they are planted; entire trees will be edible, not only their fruit; and even non-fruit-bearing trees will bear fruit. The Talmud describes the Messianic era as a time when the earth will produce delicacies and silk clothing, when wheat stalks will tower like palm trees and grains of wheat will grow as large as two kidneys of a large ox (Ketubot 111b).

On the other hand, Maimonides asserts that the nature of the world will not change in the Messianic era. His opinion is based on an opinion in the Talmud as well.

What would Maimonides do with other statements in our tradition that imply a miraculous time? An even greater difficulty with Maimonides’s assertion is posed by the fundamental belief that in the Messianic era the dead will come back to life. Maimonides lists this belief as one of the thirteen principles of faith. How can we believe that the dead will come to life, if the nature of the world will not change when the Messiah comes?

This suggests that Maimonides sees the Messianic age as consisting of two eras. In the first era, immediately following the coming of the Messiah, the world will remain in its natural state. The Messiah will not be accepted based on whether or not he performs supernatural feats, but based on whether or not he brings peace to the world, gathers the Jews to the Land of Israel, and rebuilds the Holy Temple.

Initially, there will be no change in the world order other than its readiness to accept messianic rule. All the nations of the world will strive to create a new world order in which there will be no more wars or conflicts. Jealousy, hatred, greed and political strife (of the negative kind) will disappear and all human beings will strive only for good, kindness and peace.  There will be great advances in technology allowing a high standard of living for everyone. Food will be plentiful and cheap. However the focus of human aspiration will be the pursuit of the “knowledge of G–d.” People will become less materialistic and more spiritual.

In a later era, the era of the resurrection of the dead, the nature of the world will indeed change. Only then will all the supernatural phenomena prophesied in the Bible and by our sages occur.

Source: Likutei Sichot, vol. 27, p. 191 ff.