Place crumbled brownies in a bowl. Pour rum over brownies, let soak for 1 minute. Stir until mixture forms a paste-like consistency.
Using a 3cm scoop or melon baller, scoop out balls of chocolate-rum mixture and transfer each to the plate lined with plastic wrap. Place in freezer for 15 to 20 minutes or until firm enough to roll. Remove from freezer, roll each into a smooth ball; return plate to freezer.
Melt 85g chocolate in a bowl set over a saucepan of simmering water, stirring frequently and scraping down the sides with a rubber spatula to avoid scorching. Remove from heat. Add extra chocolate; stir until chocolate is melted and combined.
Remove chocolate balls from freezer; place one ball in melted chocolate and gently move around with a fork to coat completely. Transfer to a tray lined with a silicone baking mat or baking paper. Repeat to make remaining rum balls.