Place the sugar, water, and lemon zest in a small saucepan. Bring to a boil, remove from the heat, and pour through a fine strainer to remove solids. Let cool and stir in lemon juice.
You won’t use the entire amount for this recipe; store the extra in the fridge in a sealed container for a few weeks.
Make the cocktails:
Place the blackberries and 2 Tbsp water in a blender and blend on high until smooth. If needed, add more water 1 Tbsp at a time to get it moving through the blades. Pour through your fine sieve again to remove any remaining solids.
Combine 1 cup of the blackberry puree, 1 cup simple syrup, the lemon juice and vodka. Mix well.
When ready to serve, combine 4 oz (1/2 cup) of the concentrate with 4 oz (1/2 cup) bubbly water, mix and serve with extra blackberries on top!