1. Make thyme simple syrup by bringing the sugar, water and thyme to boil in a small saucepan. Remove syrup from heat and let steep for 20 minutes, then remove thyme and discard. Stir in fresh lemon juice and let cool completely.
2. Divide the syrup between six glasses with the blackberries. Stir, breaking up the berries a bit and letting the juice mingle with the syrup.
3. Add ice and top with sparkling wine. Garnish with fresh thyme.