Add the molasses, ground ginger, and ground cinnamon to a cocktail shaker. Stir until the molasses loosens and the spices blend evenly.
Pour in the bourbon and fill the shaker halfway with ice cubes. Shake for 20 seconds to chill and combine the mixture.
Divide the contents among four glasses filled with fresh ice cubes.
Top each glass with ginger beer and stir gently once.
Place one cinnamon stick in each glass before serving.