Blanch peaches in boiling water for 30 seconds, then transfer to ice water. Peel, pit, and chop the flesh.
Blend peach flesh until completely smooth. Strain through fine-mesh sieve if desired. Chill thoroughly.
Pour 2 oz chilled peach puree into a champagne flute.
Top slowly with 4 oz chilled Prosecco, pouring down the side of the glass.
Stir gently once or twice with a bar spoon.
Serve immediately, garnished with a peach slice or mint sprig if desired.