To make spiced butter: Beat (1 stick) room-temperature unsalted butter, 1/2 cup packed dark-brown sugar, 2 teaspoons finely grated orange zest, 3/4 teaspoon ground cinnamon, 2 teaspoons fresh peeled and grated ginger, and 1/2 teaspoon ground nutmeg with a hand or stand mixer until well combined.
For each drink: Add a heaping tablespoon spiced butter to a heatproof glass. Add 1 1/2 ounces spiced rum, and top with 6 ounces boiling water. Garnish with cinnamon stick, orange zest, and a splash of orange juice if desired.
**Spiced butter will last refrigerated in a sealed container for about 2 weeks.