Pour the maple syrup and water into a small saucepan. Set the pan over medium-high heat and bring the mixture to a boil.
Add the halved cinnamon sticks, star anise, crushed ginger, orange wheel, and rosemary sprig.
Once the liquid begins boiling, lower the heat and allow it to simmer gently for 2–3 minutes so the spices, citrus, and herbs can infuse the syrup.
Remove the saucepan from the heat and leave the syrup at room temperature until it has cooled completely.
Add the fresh lemon juice and orange juice to a cocktail shaker.
Place the orange wheel and rosemary sprig into the shaker.
Using a muddler, press and turn the orange and rosemary gently to release their juice and aromatic oils. Avoid crushing them too forcefully, as excessive muddling may introduce unwanted bitterness.
Measure 2 tablespoons of the cooled spiced maple syrup into the shaker containing the muddled citrus and rosemary.
Pour in the bourbon and add the orange bitters.
Fill the shaker approximately three-quarters full with ice. Shake vigorously for 10–12 seconds, or until the outside of the shaker becomes thoroughly chilled and frosted.
Fill a rocks glass with fresh ice cubes.
Fine-strain the shaken cocktail into the prepared glass to remove small pieces of pulp and herbs.
Top the drink with approximately 2–3 ounces of ginger beer or soda water, adjusting the quantity to preference. Stir gently to combine.
Finish with a fresh rosemary sprig and an orange wheel, either placed along the rim or floated on the surface. Serve immediately while cold and effervescent.