Add rum, orange zest, sugar, and bitters to a shaker. Muddle the ingredients until the sugar dissolves.
Add the cranberry juice, cloves, and cinnamon sticks to the shaker. Shake vigorously for 30 seconds. Allow the mixture to rest for an additional minute to allow the cloves and cinnamon to infuse the rum.
Add a handful of ice to the shaker; shake for 10 seconds.
Strain the mixture into rocks glasses containing large cubes of ice. Garnish with remaining cinnamon sticks and orange zest.