The problem seems to lie in the Western mode of thinking, wherein the Roman Church seemed to have devalued the Royal Priesthood of Believers and conflated the concept of priest with that of overseer. In the Old Testament, the priests were the only ones who could approach God. In the New Testament, through our Lord Jesus Christ, the relationship between God and men was radically altered, to the point where we now "dare" to address God as Father and pray the Lord's prayer. The problem with Protestants, aside from their fanatical insistence on Sola Scriptura, is that they have generalized too much the concept of the priesthood, to the point of conflating Royal Priesthood of the Believers with the ordained priesthood, who act in our name. I think that this distinction may be an "aha" moment for Protestants, particularly Evangelicals. Take all of the definitions that Isa and JLatimer have posted and ask the Evangelicals to consider whether their approach betrays Roman Catholic thinking patterns.
I would add a few things:
1. It is clear in the scriptures that local churches are to be headed by overseers (bishops) and that they are to be assisted by deacons. It is also clear that churches contain elders who, among other things, anoint the sick and assist the bishop. In our practice, we also have deputy Bishops or rectors as we have too few Bishops to head up each congregation of a local church. However, these deputy overseers do not detract from but re-enforce the Biblical practice.
2. It is clear that the Old Testament priests made the artificial sacrifice (the original and most important function of a priests). In the New Testament, the head of a local church is also its head priest. Now, if we all are priests, one of us must be the head one--ergo the bishop of his designated deputy, the priest. That is why the vast majority of the priestly prayers are made on behalf of the entire people of God. During the Anaphora, for example, only a few prayers by the Priest are made for himself and the deacon only and those are asking God to them to do their part as priest and deacon for all the people (I am not addressing the prayers during the Fraction, which are nothing more than the necessary actions of the Priest as prompted by the Deacon, and during Communion of the Clergy, which are necessarily private).
3. Saint Ignatius' most memorable description of a local, ontologically complete church was "One bishop, surrounded by his priests, deacons and laity." This is not fundamentally different from the New Testament description of a church that consists of a bishop, deacons and the laity. The role of the priests in the Ignatian model is nothing less than Deputy Overseer, because by the time of Saint Ignatius, the overseer of the local church was based in a city with more than one congregation and, since the bishop could not be in more than one place at any given time, he had to rely on Deputy Overseers, who like him also carried priestly functions.
I think that evangelicals may understand better if you substitute the word "pastor" for "Overseer " or bishop. You can then point out that Acts and the Pauline letters refer to churches that are in effect city churches, overseen by one pastor. The Biblical Church did not have but one such overseer. However, it does stand to reason that as the number of congregations increased, there had to be a chief pastor for all of the congregations in a given city and pastors for each congregation that were in fact deputies of the one and only Biblical overseer, called bishop in modern English.
However, the problem is not the existence of bishops/overseers, elders or pastors. The problem is that many evangelicals do not believe in the sacraments or mysteries. I would only point out to them that it would be dangerous to risk misunderstanding the plain words of our Lord:
Mark 14:22-24 "While they were eating, Jesus took bread, gave thanks and broke it, and gave it to his disciples, saying, "Take it; this is my body." Then he took the cup, gave thanks and offered it to them, and they all drank from it. "This is my blood of the covenant, which is poured out for many," he said to them."
because He also said
John 6: 53-58: "Jesus said to them, "I tell you the truth, unless you eat the flesh of the Son of Man and drink his blood, you have no life in you. Whoever eats my flesh and drinks my blood has eternal life, and I will raise him up at the last day. For my flesh is real food and my blood is real drink. Whoever eats my flesh and drinks my blood remains in me, and I in him. Just as the living Father sent me and I live because of the Father, so the one who feeds on me will live because of me. This is the bread that came down from heaven. Your forefathers ate manna and died, but he who feeds on this bread will live forever."
The bottom line: the Orthodox Church is the one who fully respects and believes in the Word of God as we do not pick and choose from different verses--unlike the Evangelicals. Our core beliefs have withstood the test of time because they did not change from the very beginning. They are based on what the Lord and His Holy Apostles said and as we certified when the accepted books of the New Testament were put together: that the four Gospels, the Acts of the Apostles and the Letters of the Apostles, as assembled in the New Testament, conform to the Apostolic teachings that were passed on faithfully from disciple to disciple.
I agree that we have nothing to prove but that we can explain quite easily, all based on the Holy Scriptures as they were understood by the earliest Church. It is up to the Evangelicals to prove that their interpretation of the Scriptures is in any way superior to the understanding of the Church.