In a small bowl, whisk together the water, lemon juice, and sugar until the sugar is dissolved and then pour into a blender.
Wash and dry fresh peaches and dice into large chunks and add to the blender.
Blend until smooth and then pour the puree over a fine-mesh strainer over a large bowl, to remove any solids. Refrigerate the puree until fully chilled, or for up to 5 days.
To prepare the bellini, place 2 tablespoons of the peach puree into a champagne glass and top with the prosecco. Gently stir and then serve immediately
Peaches: Feel free to use defrosted, frozen peaches in place of the fresh peaches. And there is NO need to peel the peaches, any part of the skin that does not process in the blender, will be caught in the strainer. Sugar: If you have really ripe, sweet peaches, feel free to omit the sugar or if the peaches are underripe or not very sweet, feel free to increase the sugar slightly. To make in advance: Prepare the peach puree up to 5 days before intending to serve. Store the puree in a sealed container in the refrigerator for up 5 days. Mix well before using for the bellinis.