What does "...bring them here and kill them in front of me" in Luke 19:27 mean?

Since you asked for an "orthodox" (small "o") understanding of this, I'm going to turn to several well-known Bible commentaries, all conveniently located here, associated with the verse in question, an BibleHub.com.

Some seem to think that it is prophetical, and fulfilled in the destruction of Jerusalem, and the slaying of the Jews, who rejected Christ.

Pulpit Commentary

Verse 27. - But those mine enemies, which would not that I should reign over them, bring hither, and slay them before me. An obvious reference to the Lord's dealings with the chosen people, and an unmistakable reference to the awful ruin and disaster which was so soon to overwhelm the city and temple and the whole nationality.

Gill's Exposition of the Entire Bible

But those mine enemies,.... Meaning particularly the Jews, who were enemies to the person of Christ, and hated and rejected him, as the King Messiah; and rebelled against him, and would not submit to his government; and were enemies to his people, and were exceeding mad against them, and persecuted them; and to his Gospel, and the distinguishing truths of it, and to his ordinances, which they rejected against themselves:

which would not that I should reign over them; see Luke 19:14

bring hither, and slay them before me; which had its accomplishment in the destruction of Jerusalem, when multitudes of them were slain with the sword, both with their own, and with their enemies;

Clarke's Commentary on the Bible

Those - enemies - bring hither - the Jews, whom I shall shortly slay by the sword of the Romans.

Others seem to believe that it refers to the ultimate fate of all who reject Christ.

People's New Testament

19:27 Those my enemies. This portrays the fate, not of church members, but of those who would not have the Lord reign over them. It embraces all the impenitent. Compare Mt 13:49 21:44 25:30:00 2Th 1:8-10.

Jamieson-Fausset-Brown Bible Commentary

1. bring hither, &c.—(Compare 1Sa 15:32, 33). Referring to the awful destruction of Jerusalem, but pointing to the final destruction of all that are found in open rebellion against Christ.

Either understanding seems to be common, just with different groups and commentators.

The verse is part of a parable that Jesus taught. The parable seems intended to teach about the judgment of God at this point. It does not suggest that human governance should even follow this example. He merely uses what could and often did happen to illustrate a point of God's supreme governance.

11 As they heard these things, he proceeded to tell a parable, because he was near to Jerusalem, and because they supposed that the kingdom of God was to appear immediately. 12 He said therefore, “A nobleman went into a far country to receive for himself a kingdom and then return. 13 Calling ten of his servants, he gave them ten minas, and said to them, ‘Engage in business until I come.’ 14 But his citizens hated him and sent a delegation after him, saying, ‘We do not want this man to reign over us.’

15 When he returned, having received the kingdom, he ordered these servants to whom he had given the money to be called to him, that he might know what they had gained by doing business. 16 The first came before him, saying, ‘Lord, your mina has made ten minas more.’ 17 And he said to him, ‘Well done, good servant![c] Because you have been faithful in a very little, you shall have authority over ten cities.’ 18 And the second came, saying, ‘Lord, your mina has made five minas.’ 19 And he said to him, ‘And you are to be over five cities.’ 20 Then another came, saying, ‘Lord, here is your mina, which I kept laid away in a handkerchief; 21 for I was afraid of you, because you are a severe man. You take what you did not deposit, and reap what you did not sow.’

22 He said to him, ‘I will condemn you with your own words, you wicked servant! You knew that I was a severe man, taking what I did not deposit and reaping what I did not sow? 23 Why then did you not put my money in the bank, and at my coming I might have collected it with interest?’ 24 And he said to those who stood by, ‘Take the mina from him, and give it to the one who has the ten minas.’ 25 And they said to him, ‘Lord, he has ten minas!’ 26 ‘I tell you that to everyone who has, more will be given, but from the one who has not, even what he has will be taken away. 27 But as for these enemies of mine, who did not want me to reign over them, bring them here and slaughter them before me.’” Luke 19:11-27 ESV

Thus, this verse has nothing to do with an individual turning the other cheek, nor does it speak to the idea of human governance at all. What it does teach is that all mankind will one day face the judgment of God. Those who have hated and rejected God will be punished.