Heat until the butter has melted, and starts to boil and foam up (this is when the liquid is evaporating).
When the butter has stopped "spluttering" (or when you start to see the butter milk solids turn light golden brown), remove the butter from the heat. Let it cool down slightly.
Add the vanilla to the rum. You can use vanilla extract or an equivalent amount of vanilla bean caviar/paste or one half of a vanilla bean. If you're using vanilla extract, you can add the vanilla at this point, OR AFTER the rum has been infused with the butter (see below). If you're using vanilla bean paste/caviar or beans, then you MUST add it at this point.
Pour the melted butter into the rum and swirl (or stir) a couple of times. Close the jar and let the rum be infused for about 30 minutes - 1 hour. Don't shake the bottle.
Place the rum in the freezer overnight (up to 48 hours).
The butter layer should separate and harden while in the freezer. When the rum has infused, remove the jar from the freezer. Using a spoon or knife, break the butter layer so it's easier to remove. DO NOT DISCARD THE BUTTER, this can be used for other recipes like brownies, or for cooking too.
Pour the remaining rum through a coffee filter to remove any other solid butter left in the alcohol. Pour the strained rum into a bottle and store in the fridge and use it for clear cocktails as you need.