Broach is primarily a verb meaning (1) to bring up (a subject),or (2) to pierce in order to draw off liquid. Brooch is a noun referring to a large decorative pin or clasp. Some dictionaries list broach as a variant spelling of brooch, and the words do have related origins (in the Middle English broche, meaning pointed tool), but they are mostly kept separate in modern usage.
Broach is also not to be confused with breach.
She wanted to reach out to friends, but it never felt like the right time to broach the subject. [Washington Post]
For fall, the standout was a little piece of Surrealism for your lapel: a “painted eye” brooch. [Los Angeles Times]
Many companies are reluctant to alarm employees – or broach the sensitive issue of only recalling expats. [Financial Times]
My reward was a small enamel brooch of the Disney figure: purple floppy hat, green coat and droopy trousers that pooled around his feet. [The Australian]