To a food processor, add the blackberries, mint leaves, and sugar. Cover and puree until smooth and combined.
Let the puree sit for 2 minutes so the flavors can marry together.
Press the puree through a fine-mesh sieve to remove the seeds and the majority of the fruit pulp. (You can store the puree in an airtight container in the fridge for 1-2 days if needed.)
Pour 1/2 ounce of the blackberry mint puree into a champagne flute.
Fill the glass with champagne or prosecco. (You'll want to pour a little, let the bubble die down, and then pour some more until the glass is full.)
Garnish with a sprig of mint, if desired, and serve immediately.