In a small saucepan, combine the blackberries, mint leaves, sugar, and water. Bring to a boil over medium heat, stirring occasionally until the sugar dissolves and the blackberries begin to break down, about 5 minutes.
Remove the saucepan from heat and let the mixture cool for about 15 minutes. Strain the mixture through a fine-mesh sieve into a bowl, pressing the solids to extract as much liquid as possible. Discard the solids.
Fill glasses halfway with ice cubes. Pour about 2 ounces of the blackberry-mint syrup into each glass.
Top each glass with Prosecco, gently stirring to combine.
Garnish with additional mint leaves and a few whole blackberries, if desired. Serve immediately.