Make the brown butter-washed rum blend: Heat the butter on low for approximately 20 to 30 minutes until brown, stirring to keep solids from forming. In a flat, nonreactive container, combine the remaining ingredients and pour the warm brown butter on top. Whisk for about 2 minutes, until emulsified. Cover and let infuse for 12 hours, then place in the freezer for another 24 hours. Strain off all frozen butter fat and keep clarified liquor blend refrigerated in a sealed container.
Make the cordial: Combine the apple cider with the sarsaparilla and cinnamon in a pan over medium heat. Simmer for 15 to 20 minutes to infuse, then stir in sugar to combine. Let cool, then strain the spices and keep the liquid refrigerated in a sealed container.
Make the cocktail: Combine all ingredients in a cobbler glass with pebble ice, then top with more ice. Garnish with desired ingredients and serve.