Something that only happens every X turns
I'm no game designer, so this is just off the top of my head: 

I'd try and tie it to a mechanic that already exists in the game and which the players can't avoid, or at least get a obvious (maybe short term) advantage from, so it's unlikely to for them to forget. 

Examples:


Each player gets 5 cards at the start of the game and must play one each turn. When everyone has no cards left, execute Brazening and everyone draws 5 new cards.
A pool of 5 times number of players tokens in it, and at the start of their turn a player must take a token and do something with it. When the pool is empty, it's time to Brazen and then refill the pool.


What kind of actions do the players take in your game? Maybe if you roughly describe the mechanics, we could find a simple may to modify them.