Fill a bowl halfway with ice water. In a small saucepan, over medium heat, dissolve the sugar in 2 cups of water. Add the ginger, bring to a boil, and cook for 8 to 10 minutes. Remove and discard the ginger. Add half the peaches and cook for 2 minutes. Transfer to the ice water for 1 minute and slip off the skins; set aside. Repeat with the remaining peaches.
Pour the syrup into a bowl to cool (you should have about 1½ cups). Place the peaches and lemon juice in a bowl and, using your hands, crush the fruit. Spoon the pulp into a strainer placed over a bowl, then press it with the back of a spoon to extract the juice (you should have at least 4 cups). Stir in the syrup and refrigerate for at least 1 hour.
Divide the mixture among 4 glasses and top each with a scoop of the ice cream.