For the ginger simple syrup:
1) Peel and grate the ginger. Add to a saucepan with the sugar and cold water. Bring to the boil and stir until the sugar dissolves.
2) Turn off the heat, cover and allow to steep for 15 minutes. Strain, discard the ginger pieces and set the syrup aside to cool.
For the peach puree:
1) Place the peaches in a blender with 120ml of the ginger simple syrup. Blend until completely pureed then place into the refrigerator to cool.
For the bellini:
1) Add ice cubes to a cocktail shaker and fill half way with the peach puree. Top off with Champagne and shake vigorously to combine.
2) Pour into Champagne glasses to serve. Repeat the process as needed.