There is one NT High Priest, who is Jesus Christ, and who is priestly precisely because He offers His Body and Blood to the Father. He commanded his Church to extend this same, singular, completed sacrifice through time ("Do this is remembrance of Me") and those who offer this Sacrifice, as he commanded, are called priests.
This was not only commanded in the NT, but prophesized in the OT:
Malachi 1:11
For from the rising of the sun even to the going down of the same my name shall be great among the Gentiles; and in every place incense shall be offered to my name, and a pure offering: for my name shall be great among the heathen, said the LORD of hosts.
This is not the Levitical priesthood, because the sacrifice will be offered "in every place" and "among the Gentiles" --- and it is a "pure" (perfect) sacrifice. This must be the one pure Sacrifice which replaced all the imperfect sacrifices of the OT, the sacrifice of Jesus Christ.
As Paul says in Hebrews, chapter nine, Christ was offered once for all time and that altar was in heaven after his death and resurrection, therefore the eating of that last meal was not a sacrifice but a remembrance, a memorial, (”Do this is remembrance of Me”) and Malachi's prophecy (3:1) sets the time at Christ's appearance.
A priesthood did not exist in the primitive Christian church. Only Christ offered his blood and flesh as a high priest and those Christians that would serve as under priests AND kings would do so after their resurrection to heaven. (Rev. 20:4)