Slice the ripe peaches and fresh ginger. In a medium saucepan, combine the sliced peaches, ginger, cane sugar, and water. Heat over medium heat until the mixture starts to simmer. Then lower to a gentle simmer for about 5 minutes until the peaches soften and the sugar dissolves. Remove from heat and let cool slightly.

Once the syrup has cooled, pour it into a blender and blend until smooth to create a peach and ginger puree. Strain this through a fine mesh strainer into a large pitcher to ensure a smooth texture without pulp.

Add the fresh lemon juice to the pitcher and pour in the remaining cold water to dilute the syrup. Stir everything together until well mixed.

To serve, fill glasses with ice cubes and pour the lemonade over the ice. Garnish each glass with fresh lemon slices as desired.