In a small bowl, mash together the butter, brown sugar, cinnamon, nutmeg, allspice, cloves, and ginger until combined.
Warm the apple cider in a small saucepan over medium heat until steaming, not boiling.
Stir in the butter mixture and let it melt, blending everything together.
Remove from heat, then add the vanilla extract and rum.
Pour into heatproof mugs and garnish with cinnamon sticks if desired.
Notes
Extra spiced butter can be made ahead and stored in the fridge for up to a week. For a non-alcoholic version, replace the rum with a splash more vanilla. Letting the cinnamon stick steep longer enhances the spice.