In a large saucepan, combine the chopped peaches, granulated sugar, 1.5 cups of water, and fine sea salt. Bring the mixture to a simmer over medium heat, stirring occasionally until the sugar dissolves.

Reduce heat to low, cover, and let the peaches cook for 15-20 minutes, or until they are very soft and easily mashable with a spoon. Remove from heat.

Carefully transfer the cooked peach mixture to a blender. Blend until completely smooth. Be cautious when blending hot liquids – ensure the lid is secure and start on low speed, gradually increasing.

Pour the blended peach syrup through a fine-mesh sieve into a clean bowl or pitcher, pressing down on the solids with a spoon or spatula to extract as much liquid as possible. Discard the pulp. Allow the syrup to cool completely to room temperature, then refrigerate until chilled (at least 1 hour).

In a large pitcher, combine the chilled peach syrup, freshly squeezed lemon juice, and 4 cups of cold water. Stir well to combine.

Taste the lemonade and adjust sweetness or tartness as desired (you can add a splash more cold water for less sweetness, or a bit more lemon juice for more tartness). Serve immediately over plenty of ice, garnished with fresh peach slices and mint sprigs if desired.