To broach a new subject. To start one in conversation. The allusion is to beer tubs. If one is flat, another must be tapped. A broach is a peg or pin, and to broach a cask is to bore a hole in the top for the vent-peg.
“I did broach this business to your highness.”
Shakespeare: Henry VIII., ii. 4.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894