In a medium saucepan, combine sliced peaches, finely sliced ginger, sugar, and 1 cup of water. Place over medium heat and bring to a gentle simmer for 5 minutes then reduce to low and cook for another 5 minutes. Remove from heat and cool slightly.

Pour the cooled peach mixture into a blender and blend on high until smooth. Strain through a fine mesh strainer into a large pitcher, pressing out the juice.

Add freshly squeezed lemon juice and 5 cups of cold water to the pitcher, stirring well. Adjust sweetness with more sugar if desired.

To serve, fill glasses with ice cubes and pour the chilled mixture over the ice. Garnish with fresh lemon slices.

Store any leftovers in an airtight container in the refrigerator for up to 3 days.