At the beginning and closing of a period of exposition and adoration of the Holy Eucharist, the Priest would use his bare hands to put the luna (container with the Host) into the monstrance.
At benediction, or the blessing at the end of a period of Exposition of the Blessed Sacrament, the Priest will “shroud” himself, or use a vestment called a humeral veil. He does this for a very significant reason. As the Priest lifts the monstrance and uses it to make the sign of the Cross, he is performing “benediction”, or giving a blessing. You will notice that the Priest will have his hands covered by the humeral veil as he moves the monstrance. This signifies that it is not the Priest who is giving the blessing, but Jesus Christ Himself, who is exposed and present in the Holy Eucharist.