Make Chinese five-spice syrup: Grind the star anise, fennel seeds, cinnamon, cloves and Szechuan peppercorns until fine. In a skillet set over medium heat, add the ground spices and toast until they begin to smoke. Remove the pan from the heat and toss the spices. Return the pan to the heat until they smoke again. Repeat 3 times. Add the simple syrup and bring to a boil. Reduce heat to low, then add the honey. Simmer for 5 minutes, then remove from the heat. Cool to room temperature and strain.
Make the cocktail: Combine the Bulleit Rye Whiskey, Luxardo Maraschino liqueur, ½ ounce of the prepared Chinese five-spice syrup, lime juice and egg white in a cocktail shaker. Shake for 20 seconds. Add ice and shake again. Rinse a coupe with Rothman & Winter Orchard apricot liqueur. Strain the contents of the shaker into the coupe and garnish with a swirl of the orange bitters. Serve.