For the blackberry shrub: In a medium saucepan, cook blackberries with water and sugar on low heat until blackberries soften. Strain and discard solids. Return juice to the pan, add balsamic vinegar and rosemary. Bring to a slow boil.
Let cool, then remove rosemary and chill shrub.
For the cocktail, squeeze lemon into a champagne flute.
Add blackberry shrub and cava, stir gently and serve.