Brown the butter. Add the butter to a small saucepan over medium-low heat. Cook until the butter foams, the milk solids turn deep golden brown and it smells distinctly nutty. Take care not to burn the milk solids.
Cool briefly. Remove from the heat and leave for about 5 minutes. You want the butter warm and liquid, rather than piping hot.
Combine with the rum. Pour the rum into a jar or heatproof container. Add all of the brown butter, including the browned milk solids. Seal and shake vigorously for about 20 seconds.
Infuse. Leave at room temperature for 1–2 hours, shaking the jar a couple of times during the first 30 minutes.
Freeze. Transfer to the freezer for at least 4 hours or overnight, until the butter has formed a completely solid layer.
Remove the butter. Break through and lift off the solid layer of fat. Don’t worry about getting every tiny piece at this stage.
Strain. Pour the rum through a fine-mesh sieve, then filter through a coffee filter. For a particularly clean result, filter it a second time.
Store. Transfer to a clean, sealed bottle or jar and refrigerate until ready to use.