Make the syrup: In a small saucepan, combine the blueberries, brown sugar, and water. Bring to a gentle simmer over medium heat and let it bubble away for 10 minutes.
Smash the blueberries with the back of a spoon or a muddler to release all that gorgeous juice, then simmer for 5 more minutes.
Strain the syrup through a fine mesh strainer into a jar, pressing to get every last drop. Stir in the vanilla extract and let it cool completely. It will keep in the fridge for up to 2 weeks.
Make the cocktail: In your glass, lightly muddle the blueberries and mint leaves just until fragrant. Do not overdo it or the mint can turn bitter.
Fill the glass with ice. Add the blueberry syrup and elderflower liqueur.
Top with the chilled Prosecco and the club soda.
Stir gently to combine, garnish with a mint sprig and a few blueberries, and enjoy!