To make the banana butter, place all of the ingredients for the banana butter in the bowl of a food processor and process until combined well. Roll the mixture into a cylinder in a piece of plastic food storage wrap and refrigerate until ready to use. Let the banana butter come to room temperature before serving.
When ready to serve, add two heaping tablespoons of banana butter to each mug. Top with two ounces of rum and fill the glass two-thirds full with boiling water.
Float a star anise or two on top of the butter. Add a cinnamon stick and stir the mixture using the stick. Serve immediately.
Buttered Rum variation: prepare the banana butter, above, omitting the banana and keeping all other ingredients as specified.
Glassware: 6 large mugs