Make the dried apple: Preheat the oven 200°F (95°C). Set a wire rack over a baking sheet.

Thinly slice the apple vertically into rounds with a mandolin or sharp knife. Arrange the apple slices on the wire rack, spacing evenly.

Bake for 60 minutes, or until the apples are crisp. Remove from the oven and let cool.

Make the hot buttered rum: In a medium saucepan, whisk together the apple cider, butter, apple butter, and brown sugar. Bring to a boil over medium-high heat and cook for 1 minute, whisking continuously.

Remove the pot from the heat and whisk in the rum. If desired, pour the drink through a fine-mesh strainer to remove any solids.

Pour the hot buttered rum into 2 glasses or mugs and garnish with dried apples and cinnamon sticks. Serve warm.