Microwave blackberries in a small bowl for 1 minute. Mash with a muddler or the back of a spoon, then strain out seeds.
Add a large ice cube to 2 double Old Fashioned glasses, then divide the blackberry puree/juice into each glass. Top off with sugar, lemon juice, and gin.
Stir and top off cocktail with sparkling water. Garnish with lemon slices, blackberries, and mint if desired.