For the elderflower syrup: Combine the elderflowers, sugar and 1/2 cup water in a medium saucepan. Bring to a boil over high heat. Cook, stirring frequently, until the sugar has dissolved. Remove from the heat. Pour into a medium heatproof bowl and add the lemon. Let cool to room temperature and strain.
For the cocktail: Fill 2 wine glasses with ice. Add 1/2 cup strained elderflower syrup and 6 mint leaves to a cocktail shaker filled with ice and shake until the liquid is ice cold. Strain the mixture into the prepared glasses. Top each with 3 ounces Prosecco and a splash of club soda. Garnish with the remaining mint sprigs and with lime rounds.
If you can't find dried elderflowers, you can substitute elderflower liqueur for the syrup.