Combine 1 cup prosecco and the sugar in a small saucepan and bring to a boil, stirring until the sugar dissolves.
Meanwhile, pour the remaining prosecco into a medium bowl. Sprinkle the gelatin on top and let stand 1 minute. Pour in the hot prosecco mixture and whisk until the gelatin dissolves. Skim off most of the foam. Divide the mixture among 6 champagne glasses and refrigerate until slightly set, about 1 hour. Push a peach slice into each one, then return the glasses to the refrigerator until the gelatin is fully set, about 3 more hours.