Add the sugar and 3/4 cup water to a small saucepan. Warm over medium heat, stirring until the sugar dissolves completely. Remove from the heat and allow the simple syrup to cool.
Add the chopped peaches to a blender. For a richer peach flavor, simmer the peaches for 5 minutes with 2 tablespoons of the simple syrup before blending.
Blend until smooth. For a smoother lemonade, strain the puree through a fine-mesh sieve to remove the pulp.
In a large pitcher, combine the peach puree, fresh lemon juice, cold water, and 1/2 cup of the cooled simple syrup.
Stir well and taste. Add additional simple syrup for more sweetness, lemon juice for extra tartness, or water to dilute if needed.
Refrigerate the lemonade for at least 30 minutes before serving.
Serve over ice and garnish with peach slices, lemon wheels, or fresh mint if desired.