In John 17:19, what is Jesus referring to specifically regarding His sanctification, and how does this then apply to us being sanctified?

Jesus' ultimate sanctification occurred at the cross. The cross was the primary reason Jesus came to earth: to die, the just for the unjust, that he might bring us to God (1 Peter 3:18).

Unlike all the members of the human race who are born to live (though ultimately they will one day die), Jesus as the God-Man was born to die. We know, of course, his death was not the end of the story; rather, it was just the beginning. After his resurrection and ascension to heaven, the Scripture tells us that Jesus

ever liveth to make intercession for [us] (Hebrews 7:25).

Jesus as the Son of God could simply not die. Along with the Father and the Holy Spirit, Jesus is eternal. His priesthood, likewise, is eternal, and it was not based on his genealogy, since Jesus came not from the priestly line according to the flesh, but from the tribe of Judah (Hebrews 7:14). His priesthood, we are told, resembles the priesthood of Melchizedek, who the book of Hebrews tells us was

Without father, without mother, without genealogy, having neither beginning of days nor end of life, but [was] made like the Son of God [and therefore] he remains a priest perpetually (v.3).

In verse 16 of the same chapter, the writer tells us Jesus became this perpetual high priest

by virtue of the power of an indestructible life (Hebrews 7:16)

Jesus as the Son of Man, however, could die if he chose to do so. When Governor Pilate foolishly said to Jesus on the day he was crucified,

"Do you not know I have the authority to release you, and I have authority to crucify you?" (John 19:10),

Jesus told Pilate,

"You would have no authority over Me, unless it had been given you from above, for this reason he who delivered Me to you has the greater sin" (v.11).

Jesus' sanctification by his Father from eternity past meant that no one could take Jesus' life from him. Of a truth, Jesus said

"I lay down my life--in order to take it up again" (John 10:17).

Jesus' willingness to lay down his life pleased his Father. The prophet Isaiah tells us,

But the LORD was pleased To crush Him, putting Him to grief; If He would render Himself as a guilt offering, He will see His offspring; He will prolong His days, And the good pleasure of the LORD will prosper in His hand (53:10).

Jesus' willingness to lay down his life for the sheep--you and me--not only demonstrated his love for sinners, but it demonstrated his love for his Father, and in turn it engendered anew the Father's love for him (see John 10:17a).

Sanctification, of course, is the process of being set apart by God for a specific purpose. The prophets, priests, judges and kings of the Old Covenant were set apart by God for a specific calling or purpose. Prophets foretold and forth-told God's will and word; priests interceded for themselves and the people through their service within Israel's sacrificial system; judges often delivered God's people from threatened extinction at the hands of their enemies; and kings reigned over the people in righteousness--that is, when their hearts were committed to God and God alone.

Today, God's will for his children is that they, too, be sanctified, in the sense that they separate themselves from sin and become useful in the service of their king. As Paul tells us in 2 Timothy,

. . . if anyone purifies himself from these things, he will be a special instrument, set apart, useful to the Master, prepared for every good work (2:21 CSB, my emphasis).

Jesus, on the other hand, did not need to separate himself from sin, since there was no sin in him (see John 8:46). However, at the cross

[God] made Him who knew no sin to be sin on our behalf, so that we might become the righteousness of God in Him (2 Corinthians 5:21).

In gratitude to God, then, for what Jesus has done on our behalf at the cross, the least we can do is to consider ourselves dead to sin and alive to God and to his good and acceptable and perfect will for us (Romans 12:2). As we submit ourselves to God, his sanctifying work in our hearts will progress, albeit sometimes in fits and starts, until one day it culminates in our glorification (see Romans 8:29-30).

In conclusion, from eternity past, Jesus was set apart by God to become the one and only Lamb of God, who taketh away the sin of the world (John 1:29, 36). In so doing, Jesus fulfilled the roles of both prophet and priest. Moreover, one day he will fulfill his roles of both judge and king: the judge of all mankind, and the King of kings and the Lord of lords,

so that at the name of Jesus EVERY KNEE WILL BOW, of those who are in heaven and on earth and under the earth, and that every tongue will confess that Jesus Christ is Lord, to the glory of God the Father (Philippians 2:10, 11).