Combine diced peaches, 1 cup of water and granulated sugar in a saucepan and bring to a boil. Reduce heat and simmer until the peaches are fragrant and sugar fully dissolved, about 10 minutes. Remove from heat and let cool, about an hour.

Blend the peach mixture in a blender until quite smooth. Strain through a fine-mesh sieve, pressing on solids with the back of a spoon to release juices. Discard any solids (you may only have as little as 1 tablespoon left).

In a large pitcher, combine the peach syrup, lemon juice and 4 cups of cold water (start with ½ cup of lemon juice and increase to your personal preferences from there).

Pour the peach lemonade into ice-filled glasses and garnish with peach and lemon slices.