Make the Rosemary Simple Syrup: Combine the sugar and water in a small saucepan and heat over medium-high heat, stirring constantly, until the sugar dissolves.
Add the rosemary sprigs to the saucepan and bring the mixture to a boil. Reduce the heat to low and let the mixture simmer for 5-10 minutes, or until rosemary flavor has infused into the syrup.
Remove the saucepan from heat and let the mixture cool to room temperature. Once the syrup has cooled, remove the rosemary sprigs and strain the syrup through a fine-mesh sieve to remove any rosemary pieces.
Transfer the syrup to an airtight container and store it in the refrigerator for up to 2 weeks.
Make the Drink: Combine bourbon, blackberry purée, lemon juice, and rosemary syrup in a cocktail shaker filled with ice.
Shake vigorously to a ten count and strain into a cocktail coupe and serve.