1Make the jasmine-infused rum: stir jasmine tea into rum in a nonreactive resealable jar. Let stand 90 minutes, then strain. Keep sealed.
2Make the blueberry syrup: Heat blueberries in a saucepan, smashing berries with a muddler or wooden spoon. Add lemon juice and water, bring to a simmer. Strain mixture into a measuring cup, add an equal amount of sugar (about 1 1/2 cups.) Let cool and keep refrigerated.
3Make the vanilla simple syrup: Add water and sugar to a small saucepan. Split and scrape vanilla bean into saucepan, adding pod. Heat gently to a simmer, stirring until sugar dissolves. Let cool. Strain and keep refrigerated.
4Make cocktail: Pour rum, lime juice, orgeat, blueberry syrup, pineapple juice, vanilla simple syrup, and allspice dram into a cocktail shaker and shake without ice.
5Pour into a chilled pilsner glass and fill with crushed ice. Vibrate crushed ice with a swizzle stick or cocktail spoon to dilute and chill the ingredients. Dash Peychaud’s bitters on top of drink and add more crushed ice to create a cone. Garnish with basil sprig.