Heat the rum in a small saucepan over medium heat until the booze is reduced by half. Watch it carefully, as it evaporates quickly.
Set a heatproof bowl over but not touching a pot of gently simmering water. Add the chocolate and butter to the bowl and wait, stirring occasionally, until they're both completely melted.
Remove the bowl from the heat (careful, it will be hot!) and stir in the walnuts, rum, confectioners’ sugar, and salt. Let cool to room temperature. (You can cover and refrigerate the cooled rum balls mixture for a few hours or even up to a few days, although you'll need to let it warm to room temperature before you proceed.)
Portion the mixture into 1-inch balls. (If it's too sticky to easily roll into balls, cover and refrigerate for about 30 minutes. If it's a little crumbly, try to roll it into balls anyways and if that doesn't work, let the mixture rest at room temperature for about 30 minutes and try again.)
Roll the rum balls in sprinkles of some sort, if desired. Place the rum balls in a parchment-lined container with a lid.
Cover and refrigerate until ready to serve. The taste of the rum will be subtle at first yet will deepen with time. We found the taste after exerting patience for a few days to be perfect.