From SP (which always finds a reference to Christ):
The branch is Christ:
Isa 11:1 And there shall come forth a rod out of the stem of Jesse, and a Branch shall grow out of his roots:
The primary usage in scripture of 'aph' is not 'nose', but 'anger', translated more than ten times as many places as 'nose'.
As a prophetic metaphor, they WILL put Christ to their anger. This is fulfilled as they shout "Crucify him!"
The context for this is set in the beginning of the verse with "Have you seen, O Son of Man?"
When Jesus claims to be the Son of Man, he is suggesting that they will crucify him in their anger from this passage, among other things from other prophetic shadows.