To make simple syrup for the Bellini, combine the sugar and water in a small saucepan. Bring to a boil and simmer until the sugar has completely dissolved. Remove from the heat and allow to cool completely. You can make the simple syrup days in advance.
Place the peaches in a blender carafe with 1/4 cup of the simple syrup mixture. Puree until completely smooth.
To serve, pour the puree about a quarter of the way up a champagne glass and top it off with Prosecco and garnish with a sprig of mint. Serve alongside Italian Eggs Benedict.