How do the shepherd teaching and dedication-feast exchange connect?

Jesus describes a sheepfold: someone climbing in another way instead of the door is a thief and robber; the shepherd enters through the door. The porter opens, the sheep hear his voice, and he calls his own by name and leads them out. He goes before them and they follow his known voice, but flee a stranger’s unknown voice. The hearers do not understand the proverb. Jesus identifies Himself as the sheep’s door. Others who have come are thieves and robbers whom the sheep did not hear. Whoever enters by Him will be saved, go in and out and find pasture. The thief comes to steal, kill and destroy; He comes for abundant life. He is the good shepherd who gives his life for the sheep. A hireling, not owning the sheep and without care for them, flees the approaching wolf, which catches and scatters them. Jesus knows His own and is known by them, as the Father knows Him and He knows the Father. He lays down His life for His sheep and must bring others not of this fold, who will hear His voice, so there will be one fold and shepherd. The Father loves Him because He lays down His life to take it again. No one takes it from Him: He has power to lay it down and take it up, under His Father’s commandment.

These words divide the Jews again. Many call Him mad and possessed and ask why others listen; others ask whether such words belong to someone possessed and whether a devil can open blind eyes. At Jerusalem’s dedication feast in winter Jesus walks in Solomon’s porch in the temple. The Jews gather around Him, ask how long He keeps their souls in suspense and request a plain declaration if He is Christ. He says He speaks and they do not believe; His works in His Father’s name testify to Him. Their unbelief reflects not being His sheep. His sheep hear His voice, are known by Him and follow. He gives them everlasting life, they will not perish forever and no man will pluck them from His hand. What His Father gives Him is greater than all, and no one can snatch them from His Father’s hand. He says He and the Father are one; the edition explains one divine nature with two distinct persons.

The Jews take up stones. Jesus asks for which good work from the Father they stone Him. They answer that it is for blasphemy, not a good work, because as a man He makes Himself God. He asks about their law’s saying that they are gods: if it calls those receiving God’s word gods and scripture cannot be broken, why accuse the Father’s sanctified and sent one of blasphemy for saying He is God’s Son? If He does not do His Father’s works, they should not believe Him; if He does, they should believe the works even if not Him, so they may know and believe the Father is in Him and He in the Father. They seek to take Him, but He escapes their hands. He goes beyond Jordan to where John first baptized and stays. Many come, remark that John did no sign but said true things about Him, and believe.