Combine peeled, chopped peaches, granulated sugar, and 240 millilitres of water in a small saucepan. Bring to a gentle simmer over medium heat.
Allow the mixture to cook for 5–7 minutes, stirring occasionally, until the peaches become soft and the sugar dissolves completely.
Remove saucepan from the heat and let the mixture cool slightly. Transfer to a blender and purée until completely smooth. If desired, strain through a fine mesh sieve into a large pitcher for a smoother beverage.
Stir in the fresh lemon juice and the remaining cold water (start with 750–1,000 millilitres, adjusting to taste). Mix thoroughly until well blended.
Refrigerate until fully chilled. Serve over ice in glasses, garnishing with peach slices or lemon wedges if desired.
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