In a medium bowl, cream the unsalted butter and dark brown sugar with a spoon until smooth and light, 1–2 minutes.
Stir in the ground cinnamon, ground nutmeg, ground cloves, ground allspice, and fine salt until a uniform spiced batter forms. Cover and refrigerate while you set up, or keep at room temperature if using immediately.
For each drink, spoon 1 tablespoon of the spiced batter into a heatproof mug and add 2 ounces dark rum.
Pour in 4 ounces hot water, just off the boil, and stir until the batter fully dissolves and the drink turns glossy and aromatic, 10–15 seconds.
Garnish with a cinnamon stick and a light grating of whole nutmeg. Serve hot.